Question

Find the value of $\underset{x\to 0}{lim}120.\frac{tanx-x-x^3}{x^3}$

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Answer 1

We will sove this problem using the expansion of tanx

$tanx=x+\frac{x^3}{3}+\frac{2x^5}{15}+....$

$\Rightarrow limit=\underset{x\to 0}{lim}120\times \left(\frac{x+\frac{x^3}{3}+\frac{2x^5}{15}....-x-x^3}{x^3}\right)$

$=\underset{x\to 0}{lim}120\times \left(\frac{\frac{-2x^3}{3}+\frac{2x^5}{15}....}{x^3}\right)$

$=120\underset{x\to 0}{lim}(\frac{-2}{3}+\frac{2}{15}{x}^2.....)$

$=120\times \left(\frac{-2}{3}\right)$

$=-80$