Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 2

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$f (x)$ is continuous at $x = 0$ , then which of the following are always true?

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Important Questions on Continuity and Differentiability

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 4

The number of points where

f (x) = [\sin x + \cos x]

(where

[]

denotes the greatest integer function),

x \in (0, 2 π)

is not continuous is

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 9

Let

f (x) = [\begin{matrix} a \sin^{2 n} x & for x > 0 and n \to \infty \\ b \cos^{2 m} x - 1 & for x < 0 and m \to \infty \end{matrix}

then-

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 3

Indicate all correct alternatives if,

f (x) = \frac{x}{2} - 1,

on the interval

[0, π]

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 5

If

f (x) = \cos [\frac{π}{x}] \cos (\frac{π}{2} (x - 1));

where

[x]

is the greatest integer of

x,

then

f (x)

is continuous at-

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 6

Given

f (x) = [\begin{matrix} 3 - [\cot^{- 1} (\frac{2 x^{3} - 3}{x^{2}})] & for x > 0 \\ \{x^{2}\} \cos (e^{1 / x}) & for x < 0 \end{matrix}

, where

() & []

denotes the fractional part and the integral part functions respectively, then which of the following statement does not hold good -

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 7

Let

f

be a continuous function on

R

. If

f (1 / 4^{n}) = (\sin e^{n}) e^{- n^{2}} + \frac{n^{2}}{n^{2} + 1}

, then

f (0)

is-

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 8

Given $f (x) = b ({[x]}^{2} + [x]) + 1$ for $x \geq - 1$ $& f (x) = \sin (π (x + a))$ for $x < - 1$

where $[x]$ denotes the integral part of $x$ , then for what values of $a, b$ the function is continuous at $x = - 1$ ?

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-2>Q 10

The value(s) of

x

for which

f (x) = \frac{e^{\sin x}}{4 - \sqrt{x^{2} - 9}}

is continuous, is (are)-