Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 3

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A function $f$ is said to be a rational function, if $f (x) = \frac{g (x)}{h (x)}$ , where $g (x)$ and $h (x)$ are polynomials such that $h (x) \neq 0$ , then

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Important Questions on Limits

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 4

The value of

\lim_{x \to 0^{+}} \frac{\tan x^{\frac{1}{5}}}{{(\tan^{- 1} \sqrt{x})}^{2}} \frac{\ln (1 + 5 x)}{(e^{3 \sqrt[5]{x}} - 1)}

is

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 5

The value of

\lim_{x \to 0^{+}} {(\frac{1 - \cos (\sin^{2} x)}{x^{2}})}^{\frac{\log_{e} (1 - 2 x^{2})}{\sin^{2} x}}

is

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 6

If

\lim_{x \to 0} [\frac{α x e^{x} - β \log_{e} (1 + x) + γ x^{2} e^{- x}}{x \sin^{2} x}] = 10, α, β, γ \in R

, then the value of

α + β + γ

is __________.

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 7

If

\underset{x \to 0}{l i m} \frac{f (x)}{x^{2}} = a

and

\underset{x \to 0}{l i m} \frac{f (1 - c o s x)}{g (x) {s i n}^{2} x} = b

(where

b \neq 0

), then

\underset{x \to 0}{l i m} \frac{g (1 - c o s 2 x)}{x^{4}}

is

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 8

If

f (x) = 0

be a quadratic equation that

f (- π) = f (π) = 0

and

f (\frac{π}{2}) = - \frac{3 π^{2}}{4},

then

\underset{x \to - π}{l i m} \frac{f (x)}{s i n (s i n x)}

is equal to:

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 9

For

n \in N,

let

f_{n} (x) = \tan \frac{x}{2} (1 + \sec x) (1 + \sec 2 x) (1 + \sec 4 x) . . . . . . . (1 + \sec 2^{n} x)

then

\lim_{x \to 0} \frac{f_{n} (x)}{2 x}

is equal to:

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 10

If

f (x) = \frac{\sin (e^{x - 2} - 1)}{\log (x - 1)},

then

\lim_{x \to 2} f (x)

is given by

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Mathematics Crash Course JEE Main>Chapter 16 - Limits>Level 3>Q 11

If

a_{n}

and

b_{n}

are positive integers and

n \in N

, and

a_{n} + \sqrt{2} b_{n} = (2 + \sqrt{2})^{n}

, then the value of

\lim_{n \to \infty} (\frac{a_{n}}{b_{n}})

is: