Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 34

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Let $α, β$ be the roots of equation $a x^{2} + b x + c = 0,$ where $1 < α < β$ and $\lim_{x \to x_{0}} \frac{|a x^{2} + b x + c|}{a x^{2} + b x + c} = 1,$ then which of the following statements is correct

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Important Questions on Limits

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 36

Given a real valued function

f

such that

f (x) = \{\begin{cases} \frac{\tan^{2} [x]}{(x^{2} - [x]^{2})}, & x > 0 \\ 1, & x = 0 \\ \sqrt{{x} \cot {x}}, & x < 0 \end{cases}

where

[\cdot]

represents greatest integer function and

{\cdot}

represents fractional part function, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 37

If

f (x) = \frac{\sqrt{x^{2} + 2}}{3 x - 6},

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 38

If

\lim_{x \to 0} \frac{\sin 2 x + a \sin x}{x^{3}} = p (finite),

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 39

\lim_{x \to \infty} \frac{(a x + 1)^{n}}{x^{n} + A}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 40

If

l = \lim_{x \to \infty} (\sin \sqrt{x + 1} - \sin \sqrt{x})

and

m = \lim_{x \to - \infty} [\sin \sqrt{x + 1} - \sin \sqrt{x}],

where

[.]

denotes the greatest integer function, then :

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 41

If

f (x) = | x |^{\sin x},

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 42

If

\lim_{x \to 0} {(\cos x + a \sin b x)}^{\frac{1}{x}} = e^{2},

then the possible values of

a & b

are

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Alpha Question Bank for Engineering: Mathematics>Chapter 25 - Limits>Exercise-2>Q 43

If

\lim_{x \to 0} {(1 + a x + b x^{2})}^{\frac{2}{x}} = e^{3},

then possible values of

a

and

b

is/are