---
title: "List of limits"
chunk: 2/3
source: "https://en.wikipedia.org/wiki/List_of_limits"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T08:15:28.347754+00:00"
instance: "kb-cron"
---

If 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        
          lim
          
            x
            →
            c
          
        
        h
        (
        x
        )
        =
        L
      
    
    {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L}
  
 and 
  
    
      
        f
        (
        x
        )
        ≤
        g
        (
        x
        )
        ≤
        h
        (
        x
        )
      
    
    {\displaystyle f(x)\leq g(x)\leq h(x)}
  
 for all x in an open interval that contains c, except possibly c itself,

  
    
      
        
          lim
          
            x
            →
            c
          
        
        g
        (
        x
        )
        =
        L
        .
      
    
    {\displaystyle \lim _{x\to c}g(x)=L.}
  
 This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c.

== Polynomials and functions of the form xa ==

  
    
      
        
          lim
          
            x
            →
            c
          
        
        a
        =
        a
      
    
    {\displaystyle \lim _{x\to c}a=a}
  

=== Polynomials in x ===

  
    
      
        
          lim
          
            x
            →
            c
          
        
        x
        =
        c
      
    
    {\displaystyle \lim _{x\to c}x=c}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        (
        a
        x
        +
        b
        )
        =
        a
        c
        +
        b
      
    
    {\displaystyle \lim _{x\to c}(ax+b)=ac+b}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          x
          
            n
          
        
        =
        
          c
          
            n
          
        
      
    
    {\displaystyle \lim _{x\to c}x^{n}=c^{n}}
  
 if n is a positive integer

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        x
        
          /
        
        a
        =
        
          
            {
            
              
                
                  ∞
                  ,
                
                
                  a
                  >
                  0
                
              
              
                
                  
                    does not exist
                  
                  ,
                
                
                  a
                  =
                  0
                
              
              
                
                  −
                  ∞
                  ,
                
                
                  a
                  <
                  0
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to \infty }x/a={\begin{cases}\infty ,&a>0\\{\text{does not exist}},&a=0\\-\infty ,&a<0\end{cases}}}
  

In general, if 
  
    
      
        p
        (
        x
        )
      
    
    {\displaystyle p(x)}
  
 is a polynomial then, by the continuity of polynomials, 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        p
        (
        x
        )
        =
        p
        (
        c
        )
      
    
    {\displaystyle \lim _{x\to c}p(x)=p(c)}
  
 This is also true for rational functions, as they are continuous on their domains.

=== Functions of the form xa ===

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          x
          
            a
          
        
        =
        
          c
          
            a
          
        
        .
      
    
    {\displaystyle \lim _{x\to c}x^{a}=c^{a}.}
  
 In particular,

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          x
          
            a
          
        
        =
        
          
            {
            
              
                
                  ∞
                  ,
                
                
                  a
                  >
                  0
                
              
              
                
                  1
                  ,
                
                
                  a
                  =
                  0
                
              
              
                
                  0
                  ,
                
                
                  a
                  <
                  0
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to \infty }x^{a}={\begin{cases}\infty ,&a>0\\1,&a=0\\0,&a<0\end{cases}}}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          x
          
            1
            
              /
            
            a
          
        
        =
        
          c
          
            1
            
              /
            
            a
          
        
      
    
    {\displaystyle \lim _{x\to c}x^{1/a}=c^{1/a}}
  
. In particular,

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          x
          
            1
            
              /
            
            a
          
        
        =
        
          lim
          
            x
            →
            ∞
          
        
        
          
            x
            
              a
            
          
        
        =
        ∞
        
           for any 
        
        a
        >
        0
      
    
    {\displaystyle \lim _{x\to \infty }x^{1/a}=\lim _{x\to \infty }{\sqrt[{a}]{x}}=\infty {\text{ for any }}a>0}
  

  
    
      
        
          lim
          
            x
            →
            
              0
              
                +
              
            
          
        
        
          x
          
            −
            n
          
        
        =
        
          lim
          
            x
            →
            
              0
              
                +
              
            
          
        
        
          
            1
            
              x
              
                n
              
            
          
        
        =
        +
        ∞
      
    
    {\displaystyle \lim _{x\to 0^{+}}x^{-n}=\lim _{x\to 0^{+}}{\frac {1}{x^{n}}}=+\infty }
  

  
    
      
        
          lim
          
            x
            →
            
              0
              
                −
              
            
          
        
        
          x
          
            −
            n
          
        
        =
        
          lim
          
            x
            →
            
              0
              
                −
              
            
          
        
        
          
            1
            
              x
              
                n
              
            
          
        
        =
        
          
            {
            
              
                
                  −
                  ∞
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     is odd
                  
                
              
              
                
                  +
                  ∞
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     is even
                  
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to 0^{-}}x^{-n}=\lim _{x\to 0^{-}}{\frac {1}{x^{n}}}={\begin{cases}-\infty ,&{\text{if }}n{\text{ is odd}}\\+\infty ,&{\text{if }}n{\text{ is even}}\end{cases}}}
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        a
        
          x
          
            −
            1
          
        
        =
        
          lim
          
            x
            →
            ∞
          
        
        a
        
          /
        
        x
        =
        0
        
           for any real 
        
        a
      
    
    {\displaystyle \lim _{x\to \infty }ax^{-1}=\lim _{x\to \infty }a/x=0{\text{ for any real }}a}
  

== Exponential functions ==

=== Functions of the form ag(x) ===

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          e
          
            x
          
        
        =
        
          e
          
            c
          
        
      
    
    {\displaystyle \lim _{x\to c}e^{x}=e^{c}}
  
, due to the continuity of 
  
    
      
        
          e
          
            x
          
        
      
    
    {\displaystyle e^{x}}
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          a
          
            x
          
        
        =
        
          
            {
            
              
                
                  ∞
                  ,
                
                
                  a
                  >
                  1
                
              
              
                
                  1
                  ,
                
                
                  a
                  =
                  1
                
              
              
                
                  0
                  ,
                
                
                  0
                  <
                  a
                  <
                  1
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to \infty }a^{x}={\begin{cases}\infty ,&a>1\\1,&a=1\\0,&0<a<1\end{cases}}}
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          a
          
            −
            x
          
        
        =
        
          
            {
            
              
                
                  0
                  ,
                
                
                  a
                  >
                  1
                
              
              
                
                  1
                  ,
                
                
                  a
                  =
                  1
                
              
              
                
                  ∞
                  ,
                
                
                  0
                  <
                  a
                  <
                  1
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to \infty }a^{-x}={\begin{cases}0,&a>1\\1,&a=1\\\infty ,&0<a<1\end{cases}}}
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          
            a
            
              x
            
          
        
        =
        
          lim
          
            x
            →
            ∞
          
        
        
          
            a
          
          
            1
            
              /
            
            x
          
        
        =
        
          
            {
            
              
                
                  1
                  ,
                
                
                  a
                  >
                  0
                
              
              
                
                  0
                  ,
                
                
                  a
                  =
                  0
                
              
              
                
                  
                    does not exist
                  
                  ,
                
                
                  a
                  <
                  0
                
              
            
            
          
        
      
    
    {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{a}}=\lim _{x\to \infty }{a}^{1/x}={\begin{cases}1,&a>0\\0,&a=0\\{\text{does not exist}},&a<0\end{cases}}}
  

=== Functions of the form xg(x) ===

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        
          
            x
            
              x
            
          
        
        =
        
          lim
          
            x
            →
            ∞
          
        
        
          
            x
          
          
            1
            
              /
            
            x
          
        
        =
        1
      
    
    {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{x}}=\lim _{x\to \infty }{x}^{1/x}=1}
  

=== Functions of the form f(x)g(x) ===

  
    
      
        
          lim
          
            x
            →
            +
            ∞
          
        
        
          
            (
            
              
                x
                
                  x
                  +
                  k
                
              
            
            )
          
          
            x
          
        
        =
        
          e
          
            −
            k
          
        
      
    
    {\displaystyle \lim _{x\to +\infty }\left({\frac {x}{x+k}}\right)^{x}=e^{-k}}
  

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          
            (
            
              1
              +
              x
            
            )
          
          
            
              1
              x
            
          
        
        =
        e
      
    
    {\displaystyle \lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e}
  

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          
            (
            
              1
              +
              k
              x
            
            )
          
          
            
              m
              x
            
          
        
        =
        
          e
          
            m
            k
          
        
      
    
    {\displaystyle \lim _{x\to 0}\left(1+kx\right)^{\frac {m}{x}}=e^{mk}}
  

  
    
      
        
          lim
          
            x
            →
            +
            ∞
          
        
        
          
            (
            
              1
              +
              
                
                  1
                  x
                
              
            
            )
          
          
            x
          
        
        =
        e
      
    
    {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {1}{x}}\right)^{x}=e}
  

  
    
      
        
          lim
          
            x
            →
            +
            ∞
          
        
        
          
            (
            
              1
              −
              
                
                  1
                  x
                
              
            
            )
          
          
            x
          
        
        =
        
          
            1
            e
          
        
      
    
    {\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}
  

  
    
      
        
          lim
          
            x
            →
            +
            ∞
          
        
        
          
            (
            
              1
              +
              
                
                  k
                  x
                
              
            
            )
          
          
            m
            x
          
        
        =
        
          e
          
            m
            k
          
        
      
    
    {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}
  

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          
            (
            
              1
              +
              a
              
                (
                
                  
                    e
                    
                      −
                      x
                    
                  
                  −
                  1
                
                )
              
            
            )
          
          
            −
            
              
                1
                x
              
            
          
        
        =
        
          e
          
            a
          
        
      
    
    {\displaystyle \lim _{x\to 0}\left(1+a\left({e^{-x}-1}\right)\right)^{-{\frac {1}{x}}}=e^{a}}
  
. This limit can be derived from this limit.

=== Sums, products and composites ===

  
    
      
        
          lim
          
            x
            →
            0
          
        
        x
        
          e
          
            −
            x
          
        
        =
        0
      
    
    {\displaystyle \lim _{x\to 0}xe^{-x}=0}
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        x
        
          e
          
            −
            x
          
        
        =
        0
      
    
    {\displaystyle \lim _{x\to \infty }xe^{-x}=0}
  

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          (
          
            
              
                
                  a
                  
                    x
                  
                
                −
                1
              
              x
            
          
          )
        
        =
        ln
        ⁡
        
          a
        
        ,
      
    
    {\displaystyle \lim _{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\ln {a},}
  
 for all positive a.

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          (
          
            
              
                
                  e
                  
                    x
                  
                
                −
                1
              
              x
            
          
          )
        
        =
        1
      
    
    {\displaystyle \lim _{x\to 0}\left({\frac {e^{x}-1}{x}}\right)=1}
  

  
    
      
        
          lim
          
            x
            →
            0
          
        
        
          (
          
            
              
                
                  e
                  
                    a
                    x
                  
                
                −
                1
              
              x
            
          
          )
        
        =
        a
      
    
    {\displaystyle \lim _{x\to 0}\left({\frac {e^{ax}-1}{x}}\right)=a}
  

== Logarithmic functions ==

=== Natural logarithms ===

  
    
      
        
          lim
          
            x
            →
            c
          
        
        ln
        ⁡
        
          x
        
        =
        ln
        ⁡
        c
      
    
    {\displaystyle \lim _{x\to c}\ln {x}=\ln c}
  
, due to the continuity of 
  
    
      
        ln
        ⁡
        
          x
        
      
    
    {\displaystyle \ln {x}}
  
. In particular,

  
    
      
        
          lim
          
            x
            →
            
              0
              
                +
              
            
          
        
        log
        ⁡
        x
        =
        −
        ∞
      
    
    {\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }
  

  
    
      
        
          lim
          
            x
            →
            ∞
          
        
        log
        ⁡
        x
        =
        ∞
      
    
    {\displaystyle \lim _{x\to \infty }\log x=\infty }