I check many questions about limit calculations wihtout L'Hopital rule or series representation in this website. Nearly all answers (and best answers) find the limit values without saying "if there is a limit, then it needs to be this $L$".
"Some limit questions which are solved with assumption of existence of limit"
My question: if we assume a limit exists and then if we find a number for this limit, how can we ensure that the limit does really exist?
For example, in this limit: $$\lim_{x\to0}\frac{e^x-x-1}{x^2}.$$
I also want to give a counter-example for assuming the existence of a limit won't provide the limit value that we find is the limit: $$\lim\limits_{x\to 0}\frac{x}{|x|}.$$ Here a limit does not exist. But if we assume $$\lim\limits_{x\to 0} f(x)=L$$ where $f(x)=\frac{x}{|x|}$, then we have $$\lim\limits_{x\to 0} f(x)=\lim\limits_{x\to 0} f(-x)=L.$$ Since $$f(x)+f(-x)=0$$ for all $x \ne 0$ we will have $$2L=0,$$ which gives $L=0$.