---
title: "List of limits"
chunk: 1/3
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"
---

This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x.

== Limits for general functions ==

=== Definitions of limits and related concepts ===

  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        L
      
    
    {\displaystyle \lim _{x\to c}f(x)=L}
  
 if and only if 
  
    
      
        ∀
        ε
        >
        0
         
        ∃
        δ
        >
        0
        :
        0
        <
        
          |
        
        x
        −
        c
        
          |
        
        <
        δ
        
        ⟹
        
        
          |
        
        f
        (
        x
        )
        −
        L
        
          |
        
        <
        ε
      
    
    {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon }
  
. This is the (ε, δ)-definition of limit.
The limit superior and limit inferior of a sequence are defined as 
  
    
      
        
          lim sup
          
            n
            →
            ∞
          
        
        
          x
          
            n
          
        
        =
        
          lim
          
            n
            →
            ∞
          
        
        
          (
          
            
              sup
              
                m
                ≥
                n
              
            
            
              x
              
                m
              
            
          
          )
        
      
    
    {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)}
  
 and 
  
    
      
        
          lim inf
          
            n
            →
            ∞
          
        
        
          x
          
            n
          
        
        =
        
          lim
          
            n
            →
            ∞
          
        
        
          (
          
            
              inf
              
                m
                ≥
                n
              
            
            
              x
              
                m
              
            
          
          )
        
      
    
    {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)}
  
.
A function, 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
, is said to be continuous at a point, c, if 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        f
        (
        c
        )
        .
      
    
    {\displaystyle \lim _{x\to c}f(x)=f(c).}
  

=== Operations on a single known limit ===
If 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        L
      
    
    {\displaystyle \lim _{x\to c}f(x)=L}
  
 then:

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        [
        f
        (
        x
        )
        ±
        a
        ]
        =
        L
        ±
        a
      
    
    {\displaystyle \lim _{x\to c}\,[f(x)\pm a]=L\pm a}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        a
        f
        (
        x
        )
        =
        a
        L
      
    
    {\displaystyle \lim _{x\to c}\,af(x)=aL}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          
            1
            
              f
              (
              x
              )
            
          
        
        =
        
          
            1
            L
          
        
      
    
    {\displaystyle \lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}}
  
 if L is not equal to 0.

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        f
        (
        x
        
          )
          
            n
          
        
        =
        
          L
          
            n
          
        
      
    
    {\displaystyle \lim _{x\to c}\,f(x)^{n}=L^{n}}
  
  if n  is a positive integer

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        f
        (
        x
        
          )
          
            
              1
              n
            
          
        
        =
        
          L
          
            
              1
              n
            
          
        
      
    
    {\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L^{1 \over n}}
  
 if n is a positive integer, and if n is even, then L > 0.
In general, if g(x) is continuous at L and 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        L
      
    
    {\displaystyle \lim _{x\to c}f(x)=L}
  
 then

  
    
      
        
          lim
          
            x
            →
            c
          
        
        g
        
          (
          
            f
            (
            x
            )
          
          )
        
        =
        g
        (
        L
        )
      
    
    {\displaystyle \lim _{x\to c}g\left(f(x)\right)=g(L)}
  

=== Operations on two known limits ===
If 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        
          L
          
            1
          
        
      
    
    {\displaystyle \lim _{x\to c}f(x)=L_{1}}
  
 and 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        g
        (
        x
        )
        =
        
          L
          
            2
          
        
      
    
    {\displaystyle \lim _{x\to c}g(x)=L_{2}}
  
 then:

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        [
        f
        (
        x
        )
        ±
        g
        (
        x
        )
        ]
        =
        
          L
          
            1
          
        
        ±
        
          L
          
            2
          
        
      
    
    {\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
        [
        f
        (
        x
        )
        g
        (
        x
        )
        ]
        =
        
          L
          
            1
          
        
        ⋅
        
          L
          
            2
          
        
      
    
    {\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\cdot L_{2}}
  

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          
            
              f
              (
              x
              )
            
            
              g
              (
              x
              )
            
          
        
        =
        
          
            
              L
              
                1
              
            
            
              L
              
                2
              
            
          
        
        
        
           if 
        
        
          L
          
            2
          
        
        ≠
        0
      
    
    {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0}
  

=== Limits involving derivatives or infinitesimal changes ===
In these limits, the infinitesimal change 
  
    
      
        h
      
    
    {\displaystyle h}
  
 is often denoted 
  
    
      
        Δ
        x
      
    
    {\displaystyle \Delta x}
  
 or 
  
    
      
        δ
        x
      
    
    {\displaystyle \delta x}
  
. If 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 is differentiable at 
  
    
      
        x
      
    
    {\displaystyle x}
  
, 

  
    
      
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              x
              +
              h
              )
              −
              f
              (
              x
              )
            
            h
          
        
        =
        
          f
          ′
        
        (
        x
        )
      
    
    {\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}
  
. This is the definition of the derivative. All differentiation rules can also be reframed as rules involving limits. For example, if g(x) is differentiable at x,

  
    
      
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              ∘
              g
              (
              x
              +
              h
              )
              −
              f
              ∘
              g
              (
              x
              )
            
            h
          
        
        =
        
          f
          ′
        
        [
        g
        (
        x
        )
        ]
        
          g
          ′
        
        (
        x
        )
      
    
    {\displaystyle \lim _{h\to 0}{f\circ g(x+h)-f\circ g(x) \over h}=f'[g(x)]g'(x)}
  
. This is the chain rule.

  
    
      
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              x
              +
              h
              )
              g
              (
              x
              +
              h
              )
              −
              f
              (
              x
              )
              g
              (
              x
              )
            
            h
          
        
        =
        
          f
          ′
        
        (
        x
        )
        g
        (
        x
        )
        +
        f
        (
        x
        )
        
          g
          ′
        
        (
        x
        )
      
    
    {\displaystyle \lim _{h\to 0}{f(x+h)g(x+h)-f(x)g(x) \over h}=f'(x)g(x)+f(x)g'(x)}
  
. This is the product rule.

  
    
      
        
          lim
          
            h
            →
            0
          
        
        
          
            (
            
              
                
                  f
                  (
                  x
                  +
                  h
                  )
                
                
                  f
                  (
                  x
                  )
                
              
            
            )
          
          
            1
            
              /
            
            h
          
        
        =
        exp
        ⁡
        
          (
          
            
              
                
                  f
                  ′
                
                (
                x
                )
              
              
                f
                (
                x
                )
              
            
          
          )
        
      
    
    {\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)}}\right)}
  

  
    
      
        
          lim
          
            h
            →
            0
          
        
        
          
            
              (
              
                
                  
                    f
                    (
                    
                      e
                      
                        h
                      
                    
                    x
                    )
                  
                  
                    f
                    (
                    x
                    )
                  
                
              
              )
            
            
              1
              
                /
              
              h
            
          
        
        =
        exp
        ⁡
        
          (
          
            
              
                x
                
                  f
                  ′
                
                (
                x
                )
              
              
                f
                (
                x
                )
              
            
          
          )
        
      
    
    {\displaystyle \lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)}}\right)^{1/h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}
  

If 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 and 
  
    
      
        g
        (
        x
        )
      
    
    {\displaystyle g(x)}
  
 are differentiable on an open interval containing c, except possibly c itself, and 
  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        =
        
          lim
          
            x
            →
            c
          
        
        g
        (
        x
        )
        =
        0
        
           or 
        
        ±
        ∞
      
    
    {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty }
  
, L'Hôpital's rule can be used:

  
    
      
        
          lim
          
            x
            →
            c
          
        
        
          
            
              f
              (
              x
              )
            
            
              g
              (
              x
              )
            
          
        
        =
        
          lim
          
            x
            →
            c
          
        
        
          
            
              
                f
                ′
              
              (
              x
              )
            
            
              
                g
                ′
              
              (
              x
              )
            
          
        
      
    
    {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
  

=== Inequalities ===
If 
  
    
      
        f
        (
        x
        )
        ≤
        g
        (
        x
        )
      
    
    {\displaystyle f(x)\leq g(x)}
  
 for all x in an interval that contains c, except possibly c itself, and the limit of 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 and 
  
    
      
        g
        (
        x
        )
      
    
    {\displaystyle g(x)}
  
 both exist at c, then

  
    
      
        
          lim
          
            x
            →
            c
          
        
        f
        (
        x
        )
        ≤
        
          lim
          
            x
            →
            c
          
        
        g
        (
        x
        )
      
    
    {\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)}