Find the points of discontinuity, if any, of the following function: `f(x)={(e^x-1)/((log)_e(1+2x))\ \ \ ,\ \ \ if\ x!=0 7\ \ \ ,\ \ \ if\ x=0`

Explanation

Given:



`f(x)=` `frac{e^{x}-1}{log _{(}(1+2 x)}, text { if } x neq 0 7, text { if } x=0`



We have



`lim _{x -> 0} f(x)=lim _{x -> 0} frac{e^{x}-1}{log _{e}(1+2 x)}=lim _{x -> 0} frac{(frac{e^{x}-1}{x})}{(frac{2 log _{e}(1+2 x)}{2 x})}=frac{1}{2} times frac{lim _{x -> 0}(frac{e^{x}-1}{x})}{lim _{x -> 0}(frac{log _{e}(1+2 x)}{2 x})}=frac{1}{2}`



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