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Question Number 26591 by prakash jain last updated on 27/Dec/17

$$\mathrm{Prove}$$$$\underset{x\to -1}{\lim }(1+x)\ln (1+x)=0$$

Commented by abdo imad last updated on 27/Dec/17

$$letput1+x=u\Rightarrow {lim}_{x->-1^{+}}(1+x)ln(1+x)={lim}_{u->0^{+}}ulnu=0$$

Answered by ajfour last updated on 27/Dec/17

$$L=\underset{x\to -1}{\lim }\frac{\ln (1+x)}{\left(\frac{1}{1+x}\right)}$$$$=\underset{x\to -1}{\lim }[-\frac{{(1+x)}^2}{1+x}]=-\underset{x\to -1}{\lim }(1+x)=0.$$

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