c. $limlimits_{x \to infty}(x^{2}e^{1/x}-x^{2}-x)$
c. $limlimits_{x \to infty}(x^{2}e^{1/x}-x^{2}-x)$
Answer
Explanation:
Step1: Let $t=\frac{1}{x}$
As $x\rightarrow\infty$, then $t\rightarrow0$. The limit becomes $\lim_{t\rightarrow0}\left(\frac{e^{t}-1 - t}{t^{2}}\right)$.
Step2: Use the Mac - laurin series of $e^{t}$
The Mac - laurin series of $e^{t}=1 + t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\cdots$. Substitute it into the limit: $\lim_{t\rightarrow0}\frac{(1 + t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\cdots)-1 - t}{t^{2}}$.
Step3: Simplify the expression
$\lim_{t\rightarrow0}\frac{1 + t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\cdots-1 - t}{t^{2}}=\lim_{t\rightarrow0}\frac{\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\cdots}{t^{2}}$.
Step4: Cancel out $t^{2}$
$\lim_{t\rightarrow0}\left(\frac{1}{2}+\frac{t}{6}+\cdots\right)$.
Answer:
$\frac{1}{2}$