New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Limits Continuity and Differentiability question

2022 · Shift 1 · Q20

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Limits Continuity and Differentiability
2022 · Shift 1 · Q20

Limits Continuity and Differentiability question

2022 · Shift 1 · Q20

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityNumerical+3 / −1

Let

\alpha

be a positive real number. Let

f: \mathbb{R} \rightarrow \mathbb{R}

and

g:(\alpha, \infty) \rightarrow \mathbb{R}

be the functions defined by

f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)} .

Then the value of

\lim \limits_{x \rightarrow \alpha^{+}} f(g(x))

is

Numerical answer

More from Limits Continuity and Differentiability

If $β = x \to 0 lim \frac{e ^{x^{3}} - ( 1 - x ^{3} ) ^{\frac{1}{3}} + ( ( 1 - x ^{2} ) ^{\frac{1}{2}} - 1 ) s i n x}{x s i n ^{2} x},$ then the value of $6 β$ is .2022 · Numerical
For positive integer $n$ , define $f (n) = n + \frac{16 + 5 n - 3 n ^{2}}{4 n + 3 n ^{2}} + \frac{32 + n - 3 n ^{2}}{8 n + 3 n ^{2}} + \frac{48 - 3 n - 3 n ^{2}}{12 n + 3 n ^{2}} + \dots + \frac{25 n - 7 n ^{2}}{7 n ^{2}} .$ Then, the value of $n \to \infty lim f (n)$ …2022 · MCQ
Let f : R $\to$ R be defined by $f (x) = \frac{x ^{2} - 3 x - 6}{x ^{2} + 2 x + 4}$ Then which of the following statements is (are) TRUE?2021 · Multiple correct
Let the function f : R $\to$ R be defined by f(x) = x3 $-$ x2 + (x $-$ 1)sin x and let g : R $\to$ R be an arbitrary function. Let fg : R $\to$ R be the product function defined by (fg)(x) = f(x)g(x). Then which of the following statements…2020 · Multiple correct
let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit $x \to 0^{+} lim \frac{( 1 - x ) ^{1/ x} - e ^{- 1}}{x ^{a}}$ is equal to a non-zero real number,…2020 · Numerical
Let the functions $f : (- 1, 1) \to R$ and $g : (- 1, 1) \to (- 1, 1)$ be defined by $f (x) = ∣2 x - 1∣ + ∣2 x + 1∣$ and $g (x) = x - [x]$ , where [x] denotes the greatest integer less than or equal to x. Let $f o g : (- 1, 1) \to R$ be the…2020 · Numerical
The value of the limit $x \to \frac{π}{2} lim \frac{4 2 ( s i n 3 x + s i n x )}{( 2 s i n 2 x s i n \frac{3 x}{2} + c o s \frac{5 x}{2} ) - ( 2 + 2 c o s 2 x + c o s \frac{3 x}{2} )}$ …2020 · Numerical
Let f : R $\to$ R and g : R $\to$ R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y $\in$ R. If $x \to 0 lim g (x) = 1$ , then which of the following statements is/are TRUE?2020 · Multiple correct