Limit of a function: Definition, Formulas and Examples

In this section, we will first provide the definition of the limit of a function. Then we will list all the important limit formulas. In the end, we will learn how to use them to evaluate limits.

Left-hand limit and Right-hand limit:

At first, we will understand both the left-hand side and right-hand side limits of a function. Let $f(x)$ be a function of the variable $x$. Consider a fixed real number $a$. If the variable $x$ approaches to $a$ from the left-hand side of $a$, then the limiting value of $f(x)$ is called the left-hand limit. It is denoted by $\lim\limits_{x \to a-} f(x)$ or $f(a-0)$.

Similarly, if $x$ approaches to $a$ from the right-hand side of $a$, then the limiting value of $f(x)$ is called the right-hand limit. It is denoted by $\lim\limits_{x \to a+}f(x)$ or $f(a+0)$.

Notation and Definition: (Limit of a function)

Let $f(x)$ be a function of $x$. We say $\lim\limits_{x \to a}f(x)=L$ if the following is satisfied.

$\lim\limits_{x \to a-}f(x)=\lim\limits_{x \to a+}f(x)=L$.

In other words, both the left-hand limit and right-hand limit of $f(x)$ as $x$ tends to $a$ is equal to $L$, then we have $\lim\limits_{x \to a}f(x)=L$.

Criterion for the existence of limit:

Let $f(x)$ be a function of $x$. The limit $\lim\limits_{x \to a}f(x)$ exists, if below two conditions are satisfied.

  1. Both $\lim\limits_{x \to a-} f(x)$ and $\lim\limits_{x \to a+}f(x)$ exist.
  2. $\lim\limits_{x \to a-} f(x)=\lim\limits_{x \to a+} f(x)$

Some Properties of Limits:

Let $f(x)$ and $g(x)$ be two functions of the variable $x$. We assume that both $\lim\limits_{x \to a}f(x)$ and $\lim\limits_{x \to a}g(x)$ exist. We have the following basic properties of limits.

Limit of a constant function:

Let $c$ be a constant. Then $\lim\limits_{x \to a} c=c$.

Sum Rule of Limits:

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

Difference Rule of Limits:

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

Product Rule of Limits:

$\lim\limits_{x \to a} [f(x)\cdot g(x)]$ $=\lim\limits_{x \to a}f(x)\cdot \lim\limits_{x \to a}g(x)$

Division Rule of Limits:

$\lim\limits_{x \to a} \Big[\dfrac{f(x)}{g(x)}\Big]$ $=\dfrac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$, provided that $\lim\limits_{x \to a}g(x) \neq 0$.

Standard Formulas of Limits:

We will now gather all the limit formulas. Their proofs can be found on our page “proofs of all limit formulas“.

Limit Formulas for Algebraic Functions:

1. $\,\lim\limits_{x \to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$

2. $\,\lim\limits_{x \to 0}\dfrac{(1+x)^n-1}{x}=n$

Limit Formulas for Trigonometric Functions:

1. $\, \lim\limits_{x \to 0}\dfrac{\sin x}{x}=1$

2. $\, \lim\limits_{x \to 0}\dfrac{\tan x}{x}=1$

3. $\, \lim\limits_{x \to 0}\dfrac{\sin^{-1} x}{x}=1$

4. $\, \lim\limits_{x \to 0}\dfrac{\tan^{-1} x}{x}=1$

5. $\, \lim\limits_{x \to 0}\dfrac{1-\cos x}{x}=0$

Limit Formulas for Logarithmic and Exponential Functions:

1. $\, \lim\limits_{x \to 0}\dfrac{a^x-1}{x}=\log_e a$

2. $\, \lim\limits_{x \to 0}\dfrac{e^x-1}{x}=1$

3. $\, \lim\limits_{x \to 0}\dfrac{\log_e(1+x)}{x}=1$

4. $\, \lim\limits_{x \to 0}(1+x)^{\large{\frac{1}{x}}}=e$

5. $\, \lim\limits_{x \to \infty}\big(1+\dfrac{1}{x}\big)^x=e$

6. $\, \lim\limits_{x \to \infty}\big(1+\dfrac{n}{x}\big)^x=e^n$

Solved Examples on Limits:

Let’s see how we use the above formulas to compute the limit of a function.

Example 1: Find $\lim\limits_{x \to 2}{2x+5}$

Solution:

$\lim\limits_{x \to 2}{2x+5}$

$=\lim\limits_{x \to 2} 2x + \lim\limits_{x \to 2} 5$

$=2 \cdot 2 +5$

$=4+5$

$=9 \,\,$ ans.

Example 2: Evaluate $\lim\limits_{x \to 0}\dfrac{\sin 3x}{2x}$

Solution:

$\lim\limits_{x \to 0}\dfrac{\sin 3x}{2x}$

$=\lim\limits_{x\to 0}\dfrac{\sin 3x}{3x}\cdot \dfrac{3}{2}$

$=\dfrac{3}{2}\lim\limits_{x\to 0}\dfrac{\sin 3x}{3x}$ $=L$ (say)

Let $3x=t$

Then $t \to 0$ as $x \to 0$.

$\therefore L=\dfrac{3}{2} \lim\limits_{t \to 0}\dfrac{\sin t}{t}$

$=\dfrac{3}{2} \cdot 1$

$=\dfrac{3}{2} \,\,$ ans.

Example 3: Calculate $\lim\limits_{x \to 0}\dfrac{e^{2x}-1}{x}$

Solution:

Let $2x=t$

So $t \to 0$ as $x \to 0$

$\lim\limits_{x \to 0}\dfrac{e^{2x}-1}{x}$

$=\lim\limits_{t \to 0}\dfrac{e^t-1}{t/2}$

$=2\lim\limits_{t \to 0}\dfrac{e^t-1}{t}$

$=2 \cdot 1$

$=2 \,\, $ ans.

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