Discussion Forum

Evaluate $\lim_{x \rightarrow 2}\frac{\sqrt{(x+7)}-3\sqrt{(2x-3)}}{\sqrt[3]{(x+6)}-2\sqrt[3]{(3x-5)}}$

Answer:

Explanation:

We have

L = $\lim_{x \rightarrow 2}\frac{\sqrt{(x+7)}-3\sqrt{(2x-3)}}{\sqrt[3]{(x+6)}-2\sqrt[3]{(3x-5)}}$

($\frac{0}{0}$ form)

Let x-2 = t such that when x→2, t→0. Then

L= $\lim_{t \rightarrow 0}\frac{(t+9)^{1/2}-3(2t+1)^{1/2}}{(t+8)^{1/3}-2(3t+1)^{1/3}}$

= $\frac{3}{2}\lim_{t \rightarrow 0}\frac{(1+t/9)^{1/2}-(2t+1)^{1/2}}{(1+t/8)^{1/3}-(3t+1)^{1/3}}$

= $\frac{3}{2}\lim_{t \rightarrow 0}\frac{(\frac{1}{2}\frac{t}{9})-(2t)\frac{1}{2}}{\frac{1}{3}\frac{t}{8}-(3t)\frac{1}{3}}$

= $\frac{3}{2}\frac{\frac{1}{18}-1}{\frac{1}{24}-1}$

= $\frac{34}{23}$