Subject: Calculus

Tables of Common Limits

Evaluating limits is sometimes a very tedious job especially when you're really not into it. While it is very helpful for learners to evaluate limits firsthand, it is also useful to just refer to tables for answers to common limits. Plus, it saves time and effort. The following are the tables of common limits which I hope will help us a lot through the course of calculus. There's no need for you to memorize these.

The Page at a Glance

Some ordinary limits

Simple Functions

1.\lim_{x \to k}a=a; a is constant

2.\lim_{x \to k}x=k 
3.\lim_{x \to k}ax + b=ak + b; a and b are constants
4.\lim_{x \to k}x^{r} = k^r; r > 0 and an integer

Limits involving trigonometric functions

Simple Trigonometric Functions

1.\lim_{x \to k}\sin{x}=\sin{k} 
2.\lim_{x \to k}\cos{x}=\cos{k} 
3.\lim_{x \to k}\frac{\sin{x}}{x}=1 
4.\lim_{x \to k}\frac{1-\cos{x}}{x}=0 
5.\lim_{x \to k}\frac{1-\cos{x}}{x^2}=\frac{1}{2} 

Common limits for general functions

If \lim_{x \to k}f(x)=L_1 and \lim_{x \to k}g(x)=L_2, then

1.\lim_{x \to k}[f(x)\pm g(x)]=L_1\pm L_{2} 
2.\lim_{x \to c}[f(x)\cdot g(x)]=L_1\cdot L_{2} 
3.\lim_{x \to k}\frac{f(x)}{g(x)}=L_1\over L_{2}; L\neq 0
4.\lim_{x \to k}f(x)^{n}=L_{1}^{n}; n > 0
5.\lim_{x \to k}f(x)^{\frac{1}{n}}=L_{1}^{\frac{1}{n}}; n > 0, n is even and L_{1} > 0

Limits involving logarithmic and exponential functions

Simple Functions logarithmic and exponential functions

1.\lim_{x \to 0^+}\log_{b}{x}=-\infty 
2.\lim_{x \to \infty}\log_{b}{x}=\infty 
3.\lim_{x \to -\infty }k^x=0; k is a constant
4.\lim_{x \to \infty }k^x=\infty; k is a constant

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