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  • <img alt="\mathrm {f(x)}=\left\{\begin{array}{r}\mathrm{x}, \text { when x} <0 \\ 1, \text { when x} =0 \\ \mathrm{x}^2, \text { when x} >0\end{array}\right." src="https://entrancecorner.onco

\mathrm {f(x)}=\left\{\begin{array}{r}\mathrm{x}, \text { when x} <0 \\ 1, \text { when x} =0 \\ \mathrm{x}^2, \text { when x} >0\end{array}\right.. then discuss the continuity of \mathrm {f(x)} at \mathrm {x=0}.

Option: 1

 Limit not exist at x=0 and not continuous at x=0
 


Option: 2

 limit exist at x=0 and nctontinutus at x=0
 


Option: 3

Limit exist at x=0 and continuous at x=0
 


Option: 4

 None


Answers (1)

best_answer

Let us draw the graph of \mathrm{f(x)}

\begin{aligned} &\mathrm{ \lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} x^2=0 }\\ & \mathrm{\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} x=0}\\ &\text{But f(0)=1}\\ &\text{As, }\quad\mathrm{\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}} f(x) .} \end{aligned}

Thus, limit exists at x=0. But not continuous at x=0.

 

Posted by

Gaurav

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