Limit Continuity and Differentiability 2 Question 6
6. For each $t \in \mathbf{R}$, let $[t]$ be the greatest integer less than or equal to $t$. Then,
$$ \lim _{x \rightarrow 0^{+}} x \frac{1}{x}+\frac{2}{x}+\ldots+\frac{15}{x} $$
(2018 Main)
(a) is equal to 0
(b) is equal to 15
(c) is equal to 120
(d) does not exist (in $\mathbf{R}$ )
Show Answer
Answer:
Correct Answer: 6. (b)
Solution:
- $\lim _{x \rightarrow 0} \frac{\int _0^{x^{2}} \cos ^{2} t d t}{x \sin x}$
$\frac{0}{0}$ form
Applying L’Hospital’s rule, we get
$$ =\lim _{x \rightarrow 0} \frac{\cos ^{2}\left(x^{2}\right) \cdot 2 x-0}{x \cos x+\sin x}=\lim _{x \rightarrow 0} \frac{2 \cdot \cos ^{2}\left(x^{2}\right)}{\cos x+\frac{\sin x}{x}}=\frac{2}{1+1}=1 $$
