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Q. The value of $\displaystyle \lim_{x \to 2}$ $\int\limits^{{x}}_{{2}}$$\frac{3t^{2}}{x-2}dt$ is

WBJEEWBJEE 2015Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 2} \int_{2}^{x} \frac{3 t^{2}}{(x-2)} d t$
$=\frac{\displaystyle\lim _{x \rightarrow 2} \int_{2}^{x} 3 t^{2} d t}{\displaystyle\lim _{x \rightarrow 2}(x-2)}$
$=\frac{\displaystyle\lim _{x \rightarrow 2} 3 x^{2}}{1}$ [using L' Hospital's rule]
$=3 \times(2)^{2}=12$