[Math] Finding the limit of $\lim_{x\to 1} (x^2-\sqrt x)/(1-\sqrt x)$

Question

How do I evaluate
$$\lim_{x\to 1} \frac{(x^2-\sqrt x)}{(1-\sqrt x)}$$

Can someone explain the steps by steps solution to this problem?

Answer 1

Use the substitution $$\sqrt x=t\Rightarrow x^2=t; x\to 1, t\to 1$$ $$\lim_ {x\to 1} \frac{x^2-\sqrt x}{1-\sqrt x}=\lim_ {t\to 1} \frac{t^4-t}{1-t}=\lim_ {t\to 1} \frac{t(t^3-1)}{1-t}=\lim_ {t\to 1} \frac{-t(1-t)(t^2+t+1)}{1-t}=\lim_ {t\to 1}-t(t^2+t+1)=$$ $$=-3$$