Let $h$ be the function given by $h(x) = \displaystyle{\frac{x^{5}+x-2}{x^{2}-1}}$. We will investigate the behavior of both the numerator and denominator of $h(x)$ near the point where $x=1$. Let $f(x) = x^{5}+x-2$ and $g(x)=x^{2}-1$. Find the local linearizations of $f$ and $g$ at $a = 1$, and call these functions $L_f(x)$ and $L_g(x)$, respectively.
$L_f(x) =$
$L_g(x) =$

Explain why $h(x) \approx \displaystyle{\frac{L_f(x)}{L_g(x)}}$ for $x$ near $a=1$.

Evaluate the limit:
$\displaystyle{\lim_{x\rightarrow 1}\frac{L_f(x)}{L_g(x)}} =$

Use your work to make an educated guess about $\displaystyle{\lim_{x\rightarrow 1} h(x)}$.
$\displaystyle{\lim_{x\rightarrow 1} h(x)}=$

You could confirm your educated guess by inspecting the graph of $y=h(x)$.