palmantkf4u

2022-03-29

Use L'Hospital Rule to evaluate the following limits.

$\underset{x\to 0}{lim}\frac{{\text{tanh}}^{-1}x}{\mathrm{tan}\left(\pi \frac{x}{2}\right)}$

memantangti17

Beginner2022-03-30Added 13 answers

Given limit function is $\underset{x\to 0}{lim}\frac{{\text{tanh}}^{-1}x}{\mathrm{tan}\left(\pi \frac{x}{2}\right)}$

$\underset{x\to 0}{lim}\frac{{\text{tanh}}^{-1}x}{\mathrm{tan}\left(\pi \frac{x}{2}\right)}=\frac{{\text{tanh}}^{-1}\left(0\right)}{\mathrm{tan}\left(0\right)}=\frac{0}{0}$

Apply L'Hopital's Rule.

$\underset{x\to 0}{lim}\frac{{\text{tanh}}^{-1}x}{\mathrm{tan}\left(\pi \frac{x}{2}\right)}=\underset{x\to 0}{lim}\frac{2}{\pi {\mathrm{sec}}^{2}\left(\frac{\pi x}{2}\right)(1-{x}^{2})}$

$=\frac{2}{\pi}$

Apply L'Hopital's Rule.

Jeffrey Jordon

Expert2022-08-24Added 2605 answers

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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