I think , if there is an option of Limit Does Not Exist (DNE) , then definitely we should have to check whether limit exists or not.
Here, Left Hand Limit (LHL) = $\lim_{x\rightarrow 1^{-}}\frac{x^{7}-2x^{5} + 1}{x^{3}-3x^{2}+2} = \lim_{h\rightarrow 0}\frac{(1-h)^{7}-2(1-h)^{5} + 1}{(1-h)^{3}-3(1-h)^{2}+2}$
$= \lim_{h\rightarrow 0}\frac{-7(1-h)^{6}+10(1-h)^{4} }{-3(1-h)^{2}+6(1-h)}$ (Using L'H$\hat{o}$pital's Rule)
= $\frac{3}{3}$ = 1
Similarly , Right Hand Limit(RHL) =
$\lim_{x\rightarrow 1^{+}}\frac{x^{7}-2x^{5} + 1}{x^{3}-3x^{2}+2} = \lim_{h\rightarrow 0}\frac{(1+h)^{7}-2(1+h)^{5} + 1}{(1+h)^{3}-3(1+h)^{2}+2}$
$= \lim_{h\rightarrow 0}\frac{7(1+h)^{6}-10(1+h)^{4} }{3(1+h)^{2}-6(1+h)}$ (Using L'H$\hat{o}$pital's Rule)
= $\frac{-3}{-3}$ = 1
So, here LHL = RHL . It means Limit exists.