Question

$\underset{x\to 0}{\lim }\frac{{\left(a+x\right)}^2-a^2}{x}$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{{\left(a+x\right)}^2-a^2}{x}\right] \\ \mathrm{It}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}\frac{0}{0}. \\ \underset{x\to 0}{\lim }\left[\frac{a^2+x^2+2ax-a^2}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{x\left(x+2a\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[x+2a\right] \\ =0+2a \\ =2a \end{gathered}$$