If $f: R \rightarrow R$ is a differentiable function and $f(2)=6$, then $\lim _{x \rightarrow 2} \int_{6}^{f(x)} \frac{2t\,dt}{(x-2)}$ is:

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Definite Integrals

f : R \to R

is a differentiable function and

f (2) = 6

, then

lim_{x \to 2} \int_{6}^{f (x)} \frac{2 t d t}{( x - 2 )}

is:

Answer: (C)

x \to 2

the upper limit

f (x) \to f (2) = 6

, so the integral

\int_{6}^{f (x)} 2 t d t

collapses to zero while the denominator

x - 2

also vanishes, a

\frac{0}{0}

form. Differentiating top and bottom, the numerator's derivative by the Leibniz rule is

2 f (x) \cdot f^{'} (x)

and the denominator's is

1

. The limit is then

2 f (2) f^{'} (2) = 2 \times 6 \times f^{'} (2) = 12 f^{'} (2)

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Single correct option

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