Here we will list all the important limit formulas and see how to apply such formulas in practical examples.
Limit Formulas with Properties and Applications
Notation and Definition:
Let $f(x)$ be a function of $x$. The limit of $f(x)$ as $x$ goes \to $a$ ($x \to a$) is denoted by the symbol:
$\lim\limits_{x \to a} f(x)$.
We say that the limit $\lim\limits_{x \to a} f(x)$ exists if $\lim\limits_{x \to a-0} f(x)=\lim\limits_{x \to a+0} f(x)$, that is, $f(a-0)=f(a+0)$.
In other words, the above limit exists if the left-hand side limit is equal to the right-hand side limit. This way one can check whether the limit exists or not.
Binomial Theorem on Limit Formula
$\lim\limits_{x \to a} \dfrac{x^n-a^n}{x-a} =na^{n-1}$
limit Formulas for Trigonometric Functions
1. $\lim\limits_{x \to 0} \sin x =0$
2. $\lim\limits_{x \to 0} \cos x =1$
3. $\lim\limits_{x \to 0} \tan x =1$
4. $\lim\limits_{x \to 0} \dfrac{\sin x}{x}=1$
5. $\lim\limits_{x \to 0} \dfrac{\tan x}{x}=1$
6. $\lim\limits_{x \to 0} \dfrac{\sin^{-1} x}{x}=1$
7. $\lim\limits_{x \to 0} \dfrac{\tan^{-1} x}{x}=1$
8. $\lim\limits_{x \to 0} \dfrac{1-\cos x}{x}=0$
Limit Formulas for Logarithmic and Exponential Functions
1. $\lim\limits_{x \to 0} e^x =1$
2. $\lim\limits_{x \to 0} \log x =1$
3. $\lim\limits_{x \to 0} \dfrac{e^x-1}{x} =1$
4. $\lim\limits_{x \to 0} \dfrac{a^x-1}{x} =\log_e a$
5. $\lim\limits_{x \to 0} \dfrac{\log(1+x)}{x} =1$
6. $\lim\limits_{x \to 0} (1+x)^{\dfrac{1}{x}} =e$
7. $\lim\limits_{x \to \infty} (1+\dfrac{1}{x})^x =e$
8. $\lim\limits_{x \to \infty} (1+\dfrac{n}{x})^x =e^n$
Properties of Limits
Let $f(x)$ and $g(x)$ be two functions of x such that both the limits $\lim\limits_{x \to a} f(x)$ and $\lim\limits_{x \to a} g(x)$ exist.
Limit of a constant: $\lim\limits_{x \to a} c=c$ (c is a constant)
Addition Rule of limits: $\lim\limits_{x \to a}[f(x)+g(x)]=$ $\lim\limits_{x \to a}f(x)+ \lim\limits_{x \to a} g(x)$. This is also known as the sum rule of limits.
Subtraction Rule of limits: $\lim\limits_{x \to a}[f(x)-g(x)]=$ $\lim\limits_{x \to a}f(x)-\lim\limits_{x \to a} g(x)$. This is also known as the difference rule of limits.
Product Rule of limits: $\lim\limits_{x \to a}f(x)g(x)=\lim\limits_{x \to a}f(x) \lim\limits_{x \to a} g(x)$. This is also known as the multiplication rule of limits.
Quotient Rule of Derivatives: $\lim\limits_{x \to a}\dfrac{f(x)}{g(x)}=\dfrac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a} g(x)} ,$ provided that $\lim\limits_{x \to a} g(x) ne 0$. This is also known as the division rule of limits.
Examples of Limits
Now we will see some examples as an application of the above formulas.
Example 1: Evaluate $\lim\limits_{x \to 5} x^2$
Solution: $\lim\limits_{x \to 5} x^2= \lim\limits_{x \to 5} x \cdot x$
$= \lim\limits_{x \to 5} x \cdot \lim\limits_{x \to 5} x$
$= 5 \cdot 5$
$=25$ (application of product rule of limits)
Example 2: Find $\lim\limits_{x \to 1} (3x^2+4)$
Solution:
$\lim\limits_{x \to 1} (3x^2+4)=\lim\limits_{x \to 1}3x^2 + \lim\limits_{x \to 1}4$
$=3 \lim\limits_{x \to 1}x^2 + \lim\limits_{x \to 1}4$
$=3 \lim\limits_{x \to 1} x \cdot x + \lim\limits_{x \to 1} 4$
$=3 (1 \cdot 1)+4$
$=3+4=7$
Example 3: Find $\lim\limits_{x \to 0} \dfrac{\sin 2x}{x}$
Solution: Put z=2x. Then z→0 as x→0.
∴ $\lim\limits_{x \to 0} \dfrac{\sin 2x}{x}$ $=\lim\limits_{x \to 0} 2 \cdot \dfrac{\sin 2x}{2x}$
$=2\lim\limits_{z \to 0} \dfrac{\sin z}{z}$
$=2 \cdot 1=2$
FAQs
Q1: How to find all limit formulas?
Answer: All limit formulas are available at www.imathist.com