List of all Limit Formulas

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.

∴ $\lim\limits_{x \to 0} \dfrac{\sin 2x}{x}$ $=\lim\limits_{x \to 0} 2 \cdot \dfrac{\sin 2x}{2x}$

$=2\lim\limits_{z \to 0} \dfrac{\sin z}{z}$

$=2 \cdot 1=2$

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