Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 7

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Let $m$ and $n$ be two positive integers greater than $1$ . If $\lim_{α \to 0} (\frac{e^{\cos (α^{n})} - e}{α^{m}}) = - (\frac{e}{2})$ , then the value of $\frac{m}{n}$ is

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Important Questions on Limits

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 8

Let

α, β \in R

be such that

\lim_{x \to 0} \frac{x^{2} \sin (β x)}{α x - \sin x} = 1 .

Then

6 (α + β)

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 9

Let

f (x) = \frac{1 - x (1 + | 1 - x |)}{| 1 - x |} \cos (\frac{1}{1 - x})

for

x \neq 1

. Then

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 11

$\lim_{x \to 2} (\frac{\sqrt{1 - \cos \{2 (x - 2)\}}}{x - 2}) =$

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 12

Let

f : R \to [0, \infty)

be such that

\lim_{x \to 5} f (x)

exists and

\lim_{x \to 5} \frac{{(f (x))}^{2} - 9}{\sqrt{|x - 5|}} = 0

Then

\lim_{x \to 5} f (x)

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 13

If function $f (x)$ is differentiable at $x = a,$ then $\lim_{x \to a} \frac{x^{2} f (a) - a^{2} f (x)}{x - a}$ is

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 14

\lim_{x \to 0} \frac{(1 - \cos 2 x) (3 + \cos x)}{x \tan 4 x}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 15

\lim_{x \to 0} \frac{\sin (π \cos^{2} x)}{x^{2}}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-3>Q 16

\lim_{x \to 0} \frac{(1 - \cos 2 x) (3 + \cos x)}{x \tan 4 x}

is equal to