Use continuity to evaluate the limit.
I'm not sure what you mean per se, but I can show that this function is continuous at $\displaystyle x=1$
Let [tex]f(x)=e^{2x^5-2x}[/MATh].
1. $\displaystyle f(x)$ is defined at $\displaystyle x=1$
2. $\displaystyle \lim_{x\to1}f(x)\mbox{ exists }$
3. $\displaystyle \lim_{x\to1}f(x)=f(1)$.
$\displaystyle \therefore~{f(x)}$ is continiuous at $\displaystyle x=1$ by definition.
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You said "use continuity to evaluate the limit" so I suspect you are not required to prove this is continuous. A function is defined to be continuous at x= a if $\displaystyle \lim_{x\to a} f(x)= f(a)$. Sinmce this function is continuous, to evaluate the limit, you just have to evaluate the function itself: set x= 1.