Limit laws in calculus will help us to find limits of all the crazy functions that calculus can throw in your way. It will help us to find the limit of combined functions such as addition, subtraction, and multiplication etc. In this section we will be learning more about Limit laws and its applications.

Before we move into Limit laws, Let us learn Two fundamental limits

1. Limit of a Constant Function

Let $c$ be a constant, then

$lim_{x \to a}c$ = $c$

2. Limit of the Identity Function

$lim_{x \to a}x$ = $a$

If the limits $lim_{x \to a}f(x)$ and $lim_{x \to a}g(x)$ both exist, then

$lim_{x \to a}[f(x)+g(x)]$ = $lim_{x \to a}f(x) + lim_{x \to a}g(x)$

This law says that the limit of a sum is the
sum of the limits.
If the limits $lim_{x \to a}f(x)$ and $lim_{x \to a}g(x)$ both exist, then

$lim_{x \to a}[f(x)-g(x)]$ = $lim_{x \to a}f(x) - lim_{x \to a}g(x)$

This law says that the limit of a difference is the difference of the limits.
Suppose that $c$ is a constant and the limits $lim_{x \to a}f(x)$ exist, then

$lim_{x \to a}(cf(x))$ = $c(lim_{x \to a}f(x))$

The limit of constant times a function is the constant times the limit of the function.
If the limits $lim_{x \to a}f(x)$ and $lim_{x \to a}g(x)$ both exist, then

$lim_{x \to a}[f(x).g(x)]$ = $(lim_{x \to a}f(x)). (lim_{x \to a}g(x))$

The limit of a products is the products of the limits.
If the limits $lim_{x \to a}f(x)$ and $lim_{x \to a}g(x)$ both exist. And if $lim_{x \to a}g(x) \neq 0$ then

$lim_{x \to a}[\frac{f(x)}{g(x)}]$ = $\frac{lim_{x \to a}f(x)}{lim_{x \to a}g(x)}$

The limit of a quotient is the quotient of the limits.
If $n$ is any integer and the limits $lim_{x \to a}f(x)$ exist, then

$lim_{x \to a}(f(x))^{n}$ = $(lim_{x \to a}f(x))^{n}$

Limit of integer power of a function is the integer power of limit of the given function.
If $n$ is any integer and the limits $lim_{x \to a}f(x)$ exist, and limit is positive if $n$ is even then

$lim_{x \to a}(f(x))^{\frac{1}{n}}$ = $(lim_{x \to a}f(x))^{\frac{1}{n}}$

Limit of root of a function is the root of limit of the given function.
If $f(x) \leq g(x) \leq h(x)$ when $x$ is near $a$ (except possibly at $a$) and

$lim_{x \to a}f(x)$ = $lim_{x \to a}h(x)$ =$L$

then

$lim_{x \to a}g(x)$ = $L$
If $f$ is continuous at $b$ and $lim_{x \to a}g(x)$ = $b$, then $lim_{x \to a}f(g(x))$ = $f(b)$, that is

$lim_{x \to a}f(g(x))$ = $f(lim_{x \to a}g(x))$

This law says that limit symbol can be moved through a function symbol if the function is continuous and the limit exists.
The following are the examples of Limit laws

Solved Examples

Question 1: Evaluate the limit $lim_{x \to 3}[x+5]$.
Solution:
From the law of addition we have $lim_{x \to a}[f(x)+g(x)]$ = $lim_{x \to a}f(x) + lim_{x \to a}g(x)$
$lim_{x \to 3}[x+5]$ = $lim_{x \to 3}[x]$ +$lim_{x \to 3}[5]$
= $3+5$
= $8$

Question 2: Evaluate $lim_{x \to 2}[4x+2x]$
Solution:
$lim_{x \to 2}[4x+2x]$ = $lim_{x \to 2}[4x]$ + $lim_{x \to 2}[2x]$
= $4 (lim_{x \to 2}[x])$ + $2 (lim_{x \to 2}[x])$
= $4 \times 2 + 2 \times 2$
= $8 + 4$
= $12$


Question 3: Evaluate $lim_{x \to 2}[\frac{x^{3}}{5x}]$
Solution:
 
$lim_{x \to 2}[\frac{x^{3}}{5x}]$ = $\frac{lim_{x \to 2}x^{3}}{lim_{x \to 2}5x}$

= $\frac{2^{3}}{5 \times 2}$

= $\frac{8}{10}$

=$\frac{4}{5}$