Question: Let (x) = $\left\{ \begin{matrix} \sin x, & x \neq n\pi \\ 2, & x = n\pi \end{matrix} \right.\ $,...

Question

Let (x) = $\left\{ \begin{matrix} \sin x, & x \neq n\pi \\ 2, & x = n\pi \end{matrix} \right.\$ , where n ∈ I and

g (x) = $\left\{ \begin{matrix} x^{2} + 1, & x \neq 2 \\ 3, & x = 2 \end{matrix} \right.\$ , then $\lim_{x \rightarrow 0}$ g [(x)] is –

Answer 1

Solution

g [(x)] = $\left\{ \begin{array} { c c } { [ \mathrm { f } ( \mathrm { x } ) ] ^ { 2 } + 1 , } & \mathrm { x } \neq 2 \\ 3 , & \mathrm { x } = 2 \end{array} \right.$

∴ g [ (x)] = sin² x + 1, x ≠ nπ

3, x = nπ

R.H.L. = $\lim _ { h \rightarrow 0 }$ g [(0 + h)] = $\lim _ { h \rightarrow 0 }$ (sin² h + 1) = 1

L.H.L = $\lim _ { h \rightarrow 0 }$ g [(0 – h)] = $\lim _ { h \rightarrow 0 }$ (sin² h + 1) = 1

∴ $\lim _ { h \rightarrow 0 }$ g[ (x)] = 1.

Question: Let (x) = \(\left\{ \begin{matrix} \sin x, & x \neq n\pi \\ 2, & x = n\pi \end{matrix} \right.\ \),...

Solution

Question: Let (x) = \(\left\{ \begin{matrix} \sin x, & x \neq n\pi \\ 2, & x = n\pi \end{matrix} \right.\ \),...