# Best way to Understand the Limit Formula, Properties and Examples

In this article, you will learn what the limit of a function is and how it is calculated.

## What is Limit?

If y=f(x) is a function, for point x=a if f(x) takes some intermediate form then we can consider the values of the function that are near to a. when these values approach to some definite unique number as x approaches to a then the resulting number is called the limit of f(x) at x=a.

The limit is the most fundamental and important concept in calculus. It helps to find the values of a function at some point that may not be determined by any other method.

## Limit Formula

If y=f(x) is a function and the values of the function tend to some definite values when x approaches to a then the mathematically representation of limit of the function can be expressed as: The formula of limit is read as the limit of f(x) is equal to L when x approaches to a.

## Limit Properties

The properties of limit are:

### Sum:

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

According to the sum property of limits, the sum of limits of two functions is equal to the limit of their sum.

### Constant multiple:

$$\lim\limits_{x \to a}[bf(x)] \;=\; bL$$

### Product:

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

### Quotient:

$$\lim\limits_{x \to a} \frac{f(x)}{g(x)} \;=\; \frac{L}{K}$$

### Power Rule:

$$\lim\limits_{x \to a}[f(x)]^n \;=\; L^n$$

## How to Find the Limit of a Function?

If y=f(x) is a function and the values of the function tend to some definite values when x approaches to a then the limit of function can be calculated by substitution of limit value in the function.

$$\lim\limits_{x \to a}[f(x)] \;=\; L$$

See the below limits examples to understand how to evaluate limits. ## Limit Formulas

There are some important formulas of limits that are necessary to solve limits of any kind of function. These formulas are:

$$\lim\limits_{x \to 0} e^x \;=\; 1$$ $$\lim\limits_{x \to 0} \frac{e^x-1}{x} \;=\; 1$$ $$\lim\limits_{x \to 0} \frac{a^x-1}{x} \;=\; log_e a$$ $$\lim\limits_{x \to 0} \frac{log(1+x)}{x} \;=\; 1$$ $$\lim\limits_{x \to ∞} \left( 1 \;+\; \frac{1}{x} \right)^x \;=\; e$$ $$\lim\limits_{x \to 0} \left( 1 \;+\; x \right)^{\frac{1}{x}} \;=\; e$$ $$\lim\limits_{x \to ∞} \left( 1 \;+\; \frac{a}{x} \right)^x \;=\; e^a$$

## Related Formula:

For instantaneous rate of change, the derivative formula will:

1. Integral Formula
2. $$f(x) \;=\; \int_a^b g(x) dx$$

where
dx = infinitesimal displacement along x from a to b
g(x) = integrand function

3. Derivative Formula
4. $$\lim limits_(x to 0) \frac{δy}{δx} \;=\; \frac {f(x+δx)-f(x)}{(δx)}$$

where
δy = change in y
δx = change in x

## When does a Limit of a Function not Exist?

The limit of function exists only when the right and left hand side limit exist. If one of them does not exist then the limit of a function does not exist.

## How to Find a One-Sided Limit?

The one-sided limit is the value of the function when x approaches the limit from one side only. Let’s see an example to understand how a one-sided limit is calculated.

$$\text{Evaluate} \lim\limits_{x \to 3} \frac{x^2-9}{x-3}$$ $$\lim\limits_{x \to 3} \frac{x^2-9}{x-3}$$ $$\lim\limits_{x \to 3} \frac{(x+3)(x-3)}{x-3} = \lim\limits_{x \to 3}(x+3)$$ $$\lim\limits_{x \to 3} (x+3) \;=\;3\;+\;3\;=\;6$$

In this example, we only calculated the positive side limit of the given function. ### Alan Walker

Last updated: July 07, 2023

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