Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 7

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If $2^{x} + 2^{y} = 2^{x + y}$ , then $\frac{d y}{d x}$ is equal to:

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Important Questions on Limits and Derivatives

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 8

If

y = \sqrt{x + \sqrt{x + \sqrt{x +}}} . . . . . . . . \infty,

then

\frac{d y}{d x}

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 9

y = x^{x}, x > 0

, then

\frac{d y}{d x}

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 10

If

x^{m} \cdot y^{n} = (x + y)^{m + n}

, then

\frac{d y}{d x}

is -

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 13

\lim_{n \to \infty} \cos \frac{x}{2} \cos \frac{x}{4} \dots \dots \cos \frac{x}{2^{n}}

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 14

$\lim_{n \to \infty} (\tan θ + \frac{1}{2} \tan \frac{θ}{2} + \frac{1}{2^{2}} \tan \frac{θ}{2^{2}} + \dots . + \frac{1}{2^{n}} \tan \frac{θ}{2^{n}})$

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 15

If

\lim_{x \to y} \frac{\sin^{2} x - \sin^{2} y}{x^{2} - y^{2}} = g (y)

and the limiting value of

\lim_{y \to 0} g (y) = k

, then the value of

1 + k + k^{2} + k^{3}

+ \dots \dots + k^{2010}

is

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 16

Let

f (x) = \frac{x + x^{2} + \dots \dots + x^{n} - n}{x - 1}, g (x) = {(4^{n} + 5^{n})}^{\frac{1}{n}}

. and

α

and

β

are the roots of equation

\lim_{x \to 1} f (x)

= \lim_{n \to \infty} g (x)

, then the value of

\sum_{m = 0}^{\infty} {(\frac{1}{α} + \frac{1}{β})}^{m}

is

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Gamma Question Bank for Engineering Mathematics>Chapter 15 - Limits and Derivatives>Assignment>Q 17

The value of

\lim_{x \to 0} \{[\frac{2018 \sin x}{x}] + [\frac{2019 \tan x}{x}]\}

, (where

[x]

denotes the greatest integer

\leq x

) is