Question

$\underset{x\to a}{\lim }\frac{{\left(x+2\right)}^{5/2}-{\left(a+2\right)}^{5/2}}{x-a}$

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\left[\frac{{\left(x+2\right)}^{{\displaystyle \frac{5}{2}}}-{\left(a+2\right)}^{{\displaystyle \frac{5}{2}}}}{x-a}\right] \\ =\underset{x\to a}{\lim }\left[\frac{{\left(x+2\right)}^{{\displaystyle \frac{5}{2}}}-{\left(a+2\right)}^{{\displaystyle \frac{5}{2}}}}{\left(x+2\right)-\left(a+2\right)}\right] \end{gathered}$$

Let y = x + 2 and b = a + 2.

When x → a, then x + 2 ​→ a + 2.
⇒ y ​→ b

$$\begin{gathered} \underset{y\to b}{\lim }\left[\frac{y^{{\displaystyle \frac{5}{2}}}-b^{{\displaystyle \frac{5}{2}}}}{y-b}\right] \\ =\frac{5}{2}{\left(b\right)}^{\frac{5}{2}-1} \\ =\frac{5}{2}{b}^{\frac{3}{2}} \\ =\frac{5}{2}{\left(a+2\right)}^{\frac{3}{2}} \end{gathered}$$