Mathematics>Differential Calculus>Indeterminate Forms of Limits>L'Hospital's Rule

L'Hospital's Rule

IMPORTANT

L'Hospital's Rule: Overview

The topic will talk about the concept of L’Hospital rule in brief. We will learn its various theorems which are based on some indeterminate form. We will understand how to determine the limits related to this concept.

Important Questions on L'Hospital's Rule

HARD

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 1

Let $a_{1} > a_{2} > a_{3} > \dots > a_{n} > 1; p_{1} > p_{2} > p_{3} > \dots > p_{n} > 0;$ such that $p_{1} + p_{2} + p_{3} + \dots + p_{n} = 1$ . Also $F (x) = {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}^{1 / x},$ then $\lim_{x \to \infty} F (x)$ equals

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 2

If $\lim_{x \to 0} \frac{1 - cos x . cos 2 x . cos 3 x \dots \dots \dots \cos n x}{x^{2}}$ has the value equal to $253$ , find the value of $n$ is equal to (where $n \in N$ )

HARD

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 3

$\lim_{x \to \frac{π}{2}} (\frac{\sin x - {(\sin x)}^{\sin x}}{1 - \sin x + \ln (\sin x)}) =$

HARD

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 4

If $\lim_{x \to 0} \frac{x (1 + p \cos x) - q \sin x}{x^{3}} = 2$ , then the value of $p + 2 q$ equals

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 5

Evaluate: $\lim_{x \to a} \frac{\log (x - a)}{\log (e^{x} - e^{a})}$

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 6

$lim_{x \to 0} (\cos x)^{\cot^{2} x}$

HARD

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 7

If $\lim_{x \to 0} \frac{x (1 + a \cos x) - b \sin x}{x^{3}} = 1$ , then the value of $a b$ is equal to

EASY

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 8

$\lim_{n \to \infty} [\frac{1}{1 - n^{2}} + \frac{2}{1 - n^{2}} + \dots + \frac{n}{1 - n^{2}}]$ is equal to

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 9

The value of $\lim_{n \to \infty} {(1 + \tan \frac{b}{n})}^{n}$ is

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 10

$\lim_{x \to 0} x^{x}$ is equal to

HARD

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 11

The value of $\lim_{x \to \infty} {(\frac{2 x^{2} + 3}{2 x^{2} + 5})}^{8 x^{2} + 3}$ is

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 12

$\lim_{x \to 0} {(\cos x + a \sin b x)}^{\frac{a}{x}}$ is equal to

EASY

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 13

$\lim_{x \to \infty} \frac{2 x^{3} - 4 x + 7}{3 x^{3} + 5 x^{2} - 4}$ is equal to

EASY

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 14

The value of $\lim_{x \to \infty} \frac{\sqrt{1 + x^{4}} - (1 + x^{2})}{x^{2}}$ is

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 15

Evaluate:

$\lim_{x \to \infty} {(1 + x)}^{\frac{1}{x}}$

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 16

Evaluate:

$\lim_{x \to α} {(x - α)}^{(x - α)}$

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 17

The value of $\lim_{x \to \infty} {(\frac{π}{2} - \tan^{- 1} x)}^{\frac{1}{x}}$ is

EASY

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 18

$\lim_{x \to 2} [\frac{1}{x - 2} - \frac{2 (2 x - 3)}{x^{3} - 3 x^{2} + 2 x}]$ is equal to

EASY

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 19

If $f (2) = 4, f^{'} (2) = 1$ then find $\lim_{x \to 2} [\frac{x f (2) - 2 f (x)}{x - 2}]$

MEDIUM

IMPORTANT

Mathematics>Indeterminate Forms of Limits>L'Hospital's Rule> Q 20

For $x \in R, \lim_{x \to \infty} {(\frac{x - 3}{x + 2})}^{x}$ is equal to