Let $$f(x)=x1+21/x$$, if x≠$$00$$ if $$x=0$$ then $$f10$$−$$=$$

$$f10$$−$$=limx$$→$$0$$− $$f(x)$$−$$f(0)x$$−$$0$$ $$=limx$$→$$0$$− $$1+21xx$$ $$=11+2$$−$$10$$ $$=11+2$$−∞ $$=11+0$$ $$=1$$

Differentiability

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$Let f (x) = \{\begin{array}{l} \frac{x}{1 + 2^{1 / x}}, if x \neq 0 \\ 0 if x = 0 \end{array} then f^{1} (0^{-}) =$

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Solution

$\begin{array}{l} f^{1} (0^{-}) = lim_{x \to 0^{-}} \frac{f (x) - f (0)}{x - 0} \\ = lim_{x \to 0^{-}} \frac{1 + 2^{\frac{1}{x}}}{x} \\ = \frac{1}{1 + 2^{- \frac{1}{0}}} \\ = \frac{1}{1 + 2^{- \infty}} \\ = \frac{1}{1 + 0} \\ = 1 \end{array}$

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