Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 73: 53

$$\lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x} =3a^2$$

Work Step by Step

Given $$\lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x}$$ let $$f(x) = \frac{(x+a)^{3}-a^{3}}{x}$$ Since, we have $$f(0)= \frac{a^3-a^{3}}{0}=\frac{0}{0}$$ So, transform algebraically and cancel \begin{aligned}L&= \lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x} \\ &= \lim _{x \rightarrow 0} \frac{(x+a-a)((x+a)^2+a(x+a)+a^2)}{x} \\ &= \lim _{x \rightarrow 0} \frac{(x )((x+a)^2+a(x+a)+a^2)}{x} \\ &= \lim _{x \rightarrow 0}( (x+a)^2+a(x+a)+a^2) \\ &=((0+a)^2+a(0+a)+a^2)\\ &=a^2+a^2+a^2\\ &=3a^2 \end{aligned}

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