Question

Find the limit, Question attached

Answer 1

\[\lim_{x\to2}\frac{3\sqrt{x-1}-3}{x-2}\cdot\frac{3\sqrt{x-1}+3}{3\sqrt{x-1}+3}\\
\lim_{x\to2}\frac{9(x-1)-9}{(x-2)(3\sqrt{x-1}+3)}\\
9\lim_{x\to2}\frac{x-1-1}{(x-2)(3\sqrt{x-1}+3)}\\
9\lim_{x\to2}\frac{x-2}{(x-2)(3\sqrt{x-1}+3)}\\
\lim_{x\to2}\frac{9}{3\sqrt{x-1}+3}\]

Answer 2

\[\begin{align*}&\lim_{x\to0}\frac{(x+64)^{1/3}-4}{x}\cdot\frac{(x+64)^{2/3}+4(x+64)^{1/3}+16}{(x+64)^{2/3}+4(x+64)^{1/3}+16}\\\\
&\lim_{x\to0}\frac{(x+64)^{3/3}-64}{x[(x+64)^{2/3}+4(x+64)^{1/3}+16]}\\\\
&\lim_{x\to0}\frac{x}{x[(x+64)^{2/3}+4(x+64)^{1/3}+16]}\\\\
&\lim_{x\to0}\frac{1}{(x+64)^{2/3}+4(x+64)^{1/3}+16}\end{align*}\]

Answer 3

\[\begin{align*}&\lim_{x\to\infty}\frac{x^2-x^3+2x}{5x(x-2x^2)}\\\\
&\lim_{x\to\infty}\frac{x^3\left(\dfrac{1}{x}-1+\dfrac{2}{x^2}\right)}{x^3\left(\dfrac{5}{x}-10\right)}\\\\
&\lim_{x\to\infty}\frac{\dfrac{1}{x}-1+\dfrac{2}{x^2}}{\dfrac{5}{x}-10}
\end{align*}\]