JEE Main Online Test Series 2025: Free Practice Papers Here

# Limit of Indeterminate Form and Algebraic limit - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Limit of Indeterminate Form and Algebraic limit is considered one the most difficult concept.

• 94 Questions around this concept.

## Solve by difficulty

Let $\dpi{100} f : R\rightarrow R$ be a positive increasing function with $\dpi{100} \lim_{x\rightarrow \infty }\frac{f(3x)}{f(x)}=1.\; Then\; \lim_{x\rightarrow \infty }\frac{f(2x)}{f(x)}=$

Find limit  $\lim _{x \rightarrow 3} \frac{\left(x^2-4 x+3\right)}{\left(x^2-5 x+6\right.}$

Find Limit : $\lim _{x \rightarrow 2} \frac{\left(3 x^2+2 x-5\right)}{\left(x^2-1\right)}$

Find limit:  $\lim _{x \rightarrow 3} \frac{\left(x^2-5 x+6\right)}{(x-3)^3}$

The value of $\lim _{n \rightarrow \infty} \frac{1+2-3+4+5-6+\cdots \ldots+(3 n-2)+(3 n-1)-3 n}{\sqrt{2 n^{4}+4 n+3-\sqrt{n^{4}+5 n+4}}}$ is:

Apply to Amity University, Noida B.Tech Admissions 2024

$\mathrm{ If \lim _{x \rightarrow \infty}\left\{\frac{x^2+1}{x+1}-(\alpha x+\beta)\right\}=0, then\, \, \alpha \, \, \, and\, \, \beta \, \, are \, \, given\, \, by }$

Find $\lim_{\infty}(\tan x)^{\tan 2 x}$:

##### Amity University, Noida B.Tech Admissions 2024

Asia's Only University with the Highest US & UK Accreditation

Ranked #52 among universities in India by NIRF | Highest CTC 50 LPA | 100% Placements | Last Date to Apply: 20th July

The value of $\lim_{x\rightarrow 1}\mathrm{(1-x)\tan \left ( \frac{\pi\: x}{2} \right )}$ is

Is the function $\mathrm{\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}}$defined for all values of x ? Indicate the values of x for which it is defined and real. Find the limit as $\mathrm{ x \rightarrow 0.}$

JEE Main Exam's High Scoring Chapters and Topics
This free eBook covers JEE Main important chapters & topics to study just 40% of the syllabus and score up to 100% marks in the examination.

Evaluate  $\mathrm{\operatorname{Lt}_{x \rightarrow 0} \frac{x \cdot 2^x-x}{1-\cos x}}$

## Concepts Covered - 1

Limit of Indeterminate Form and Algebraic limit

Limit of Indeterminate Form and Algebraic limit

If we directly substitute x = a in f(x) while evaluating $\lim_{x\rightarrow a}f(x)$, and will get one of the seven following forms $\frac{0}{0},\frac{\infty}{\infty},\infty-\infty,1^\infty,0^0,\infty^0,\infty\times0$ then it is called indeterminate form.

For example,

$\\(i)\;\lim_{x\rightarrow 2}\frac{x^2-4}{x-2}=\frac{0}{0}\;\text{indeterminate form.}\\(ii)\;\lim_{x\rightarrow 0}\frac{\sin x}{x}=\frac{0}{0}\;\text{indeterminate form.}\\(iii)\;\lim_{x\rightarrow \pi/2}\left (\tan x \right )^{\cos x}=\infty^0\;\text{indeterminate form.}$

Note:

1. $\frac{0}{0}$ form means the numerator and denominator are both tending to 0 (AND NOT exactly 0)

Eg, $\lim_{x\rightarrow 0^+}\frac{[x]}{x}$, when we input the values of x close to 0(and x>0), then denominator is tending to 0, but numerator values are not tending to 0 BUT numerator is exactly 0. So this is not $\frac{0}{0}$ form.

2. We can convert one indeterminate form into another and vice verse.

We will divide the problems of  finding limits into five categories, which are

1. Limit of Algebraic function

2. Trigonometric Limit

3. Logarithmic limit

4. Exponential limit

5. Miscellaneous forms

One by one we will discuss all of these.

1. Limit of Algebraic function

(a) Direct Substitution Method

To find $\lim_{x\rightarrow a}f(x)$, directly substitute the value of the limit of the variable. (i.e. substitute x = a) in the expression f (x)

• If f (a) is finite then L = f (a)
• If f (a) is undefined then limit does not exist.
• If f (a) is indeterminate then this method fails.

$\\\text{(i)}\;\;\lim_{x\rightarrow 3}\left (x(x+1) \right )=3(3+1)=12\\\\\text{(ii)}\;\;\lim _{x \rightarrow 1}\left (\frac{x^{2}+1}{x+100} \right )=\frac{1+1}{1+100}=\frac{2}{101}\\\\\text{(iii)}\;\;\lim _{x \rightarrow -1}\left (1+x+x^{2}+\ldots+x^{10} \right )=1+(-1)+(-1)^{2}+\ldots+(-1)^{10}=1$

## Study it with Videos

Limit of Indeterminate Form and Algebraic limit

"Stay in the loop. Receive exam news, study resources, and expert advice!"