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  • If  is an function such that <img alt="\mathrm{\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}}" src="https://entra

If f is an function such that \mathrm{\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}} has some finite non-zero value, then

Option: 1

f is continuous and derivable at x=0


Option: 2

f is continuous but not differentiable at x=0


Option: 3

f may be discontinuous at x=0


Option: 4

None of these


Answers (1)

best_answer

Let 

\mathrm{\begin{aligned} f^{\prime}\left(0^{+}\right) & =\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=k(\text { say }) \\ f^{\prime}\left(0^{-}\right) & =\lim _{h \rightarrow 0} \frac{f(0)-f(0-h)}{h} \\ & =\lim _{h \rightarrow 0} \frac{f(0)-f(h)}{h}=-k . \end{aligned}}

since \mathrm{f^{\prime}\left(0^{+}\right) \neq f^{\prime}\left(0^{-}\right)}, but both are finite,we can say \mathrm{f(x)} is continuous at x=0 but not differentiable at x.

Posted by

Shailly goel

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