Let $$f:Rrightarrow R$$ be a continuously differentiable function such that $$f(2)=6$$ and $$f$$ . If $$int_{6}^{fleft ( x right )}4t^{3}dt=(x-2)g(x)$$ , then $$lim_{xrightarrow 2}g(x)$$ is equal to :

Help me solve this - Limit , continuity and differentiability

Let $f:R\rightarrow R$ be a continuously differentiable function such that $f(2)=6$ and $f'(2)=\frac{1}{48}$ . If $\int_{6}^{f\left ( x \right )}4t^{3}dt=(x-2)g(x)$ , then $\lim_{x\rightarrow 2}g(x)$ is equal to :

Option 1)
$36$
Option 2)
$24$
Option 3)
$12$
Option 4)
$18$

Answers (1)

A Aadil.Khan

$f:R\rightarrow R$ $f(2)=6$ and $f'(2)=\frac{1}{48}$

$\lim_{x\rightarrow 2}g(x)=\lim_{x\rightarrow 2}\frac{\int _6^{f(x)}4t^3dt}{x-2}$

$=\lim_{x\rightarrow 2}\frac{4\cdot f^{3}(x)\cdot f'(x)}{1}$

$=4f^{3}\left ( 2 \right )f'\left ( 2 \right )$

$=4\times \left ( 6 \right )^{3}\times \frac{1}{48}$

$=6\times 36\times \frac{1}{12}=18$

Option 1)

$36$

Option 2)

$24$

Option 3)

$12$

Option 4)
$18$

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Let be a continuously differentiable function such that and . If , then is equal to : Option 1)Option 2)Option 3)Option 4)

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Let $f:R\rightarrow R$ be a continuously differentiable function such that $f(2)=6$ and $f'(2)=\frac{1}{48}$ . If $\int_{6}^{f\left ( x \right )}4t^{3}dt=(x-2)g(x)$ , then $\lim_{x\rightarrow 2}g(x)$ is equal to :

Option 1)
$36$
Option 2)
$24$
Option 3)
$12$
Option 4)
$18$