Limit Continuity and Differentiability 4 Question 3
3. Let [.] denotes the greatest integer function and $f(x)=\left[\tan ^{2} x\right]$, then
(1993, 1M)
(a) $\lim _{x \rightarrow 0} f(x)$ does not exist
(b) $f(x)$ is continuous at $x=0$
(c) $f(x)$ is not differentiable at $x=0$
(d) $f^{\prime}(0)=1$
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Answer:
Correct Answer: 3. (b)
Solution:
- Given function
$$ f(x)=\begin{array}{ll} a|\pi-x|+1, & x \leq 5 \\ b|x-\pi|+3, & x>5 \end{array} $$
and it is also given that $f(x)$ is continuous at
Clearly,
$$ f(5)=a(5-\pi)+1 $$
$$ \begin{aligned} \lim _{x \rightarrow 5^{-}} f(x) & =\lim _{h \rightarrow 0}[a|\pi-(5-h)|+1] \\ & =a(5-\pi)+1 \end{aligned} $$
and $\lim _{x \rightarrow 5^{+}} f(x)=\lim _{h \rightarrow 0}[b|(5+h)-\pi|+3]$
$$ =b(5-\pi)+3 $$
$\because$ Function $f(x)$ is continuous at $x=5$.
$$ \begin{aligned} & \therefore f(5)=\lim _{x \rightarrow 5^{+}} f(x)=\lim _{x \rightarrow 5^{-}} f(x) \\ & \Rightarrow \quad a(5-\pi)+1=b(5-\pi)+3 \\ & \Rightarrow \quad(a-b)(5-\pi)=2 \\ & \Rightarrow \quad a-b=\frac{2}{5-\pi} \end{aligned} $$
