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Question: At the point x = 1, the function f (x) = \(\lim_{x \rightarrow \pi/2}\) is –...

At the point x = 1, the function f (x) = lim⁡x→π/2\lim_{x \rightarrow \pi/2} is –

A

Continuous and differentiable

B

Continuous and not differentiable

C

Discontinuous and differentiable

D

Discontinuous and not differentiable

Answer

Continuous and not differentiable

Explanation

Solution

f(1)=1−1=0f(1+)=h→lim⁡0(1+h)−1=0f(1−)=h→lim⁡→0(1−h)3−1=0}\left. \begin{array} { l } f ( 1 ) = 1 - 1 = 0 \\ f ( 1 + ) = h \xrightarrow { \lim } 0 ( 1 + h ) - 1 = 0 \\ f ( 1 - ) = h \rightarrow \lim _ { \rightarrow } 0 ( 1 - h ) ^ { 3 } - 1 = 0 \end{array} \right\} ⇒ conti. atx = 1

f '(x) = [3x2,1<x<∞1,−∞<x≤1\left[ \begin{array} { c c } 3 \mathrm { x } ^ { 2 } & , \quad 1 < \mathrm { x } < \infty \\ 1 & , \quad - \infty < \mathrm { x } \leq 1 \end{array} \right.

⇒ not diff. at x = 1