Limit Calculation

Attachment 26349

please help me with this one to

If $\displaystyle x\rightarrow 2$, then $\displaystyle 5x-10\rightarrow0$ and $\displaystyle \sqrt[3]{x-1}-1\rightarrow0$. So we have an indeterminate form, $\displaystyle \frac{0}{0}$, use l'Hôpital's rule.

The answer is 15.

A standard way to find limits of the form $\displaystyle \lim_{x\to a}\frac{f(x)-c^2}{\sqrt{f(x)}-c}$ where $\displaystyle \lim_{x\to a}f(x)=c^2$ is to multiply the numerator and the denominator by $\displaystyle \sqrt{f(x)}+c$ to get

$\displaystyle \frac{(f(x)-c^2)(\sqrt{f(x)}+c)}{(\sqrt{f(x)}-c)(\sqrt{f(x)}+c)} = \frac{(f(x)-c^2)(\sqrt{f(x)}+c)}{f(x)-c^2} = \sqrt{f(x)}+c$

where the latter expression does not have a singularity at $\displaystyle a$.

For this problem, use the same idea and the fact that

$\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)$ (*)

Equivalently, factor out 5 and represent x - 2 in the numerator as (x - 1) - 1 and then factor it according to (*).

Thank you very much now i get it :)