Answered: tan 4z lim zo tan 7z

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section4.4: Powers Of Monimials

Problem 43WE

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Question

Evaluate the following limit. Use l’Hôpital’s Rule when it is convenient and applicable.

Expert Solution

Step 1

L'Hospital rule: If

$\underset{x \to a}{l i m} \frac{f (x)}{g (x)} = \frac{0}{0} [I n d e t e r m i n a n t f o r m] T h e n \underset{x \to a}{l i m} \frac{f (x)}{g (x)} = \underset{x \to a}{l i m} \frac{\frac{d}{d x} f (x)}{\frac{d}{d x} g (x)}$

Step 2

Evaluate

$\underset{z \to 0}{l i m} \frac{\tan 4 z}{\tan 7 z}$

On applying the limit we get indeterminant (0/0) form so using L'Hospital rule,

$\begin{array}{rcl} \underset{z \to 0}{l i m} \frac{\tan 4 z}{\tan 7 z} & = & \underset{z \to 0}{l i m} \frac{\frac{d}{d z} (\tan 4 z)}{\frac{d}{d z} (\tan 7 z)} \\ = & \underset{z \to 0}{l i m} \frac{4 s e c^{2} 4 z}{7 s e c^{2} 7 z} \end{array}$

Now apply the limit,

$\begin{array}{rcl} \underset{z \to 0}{l i m} \frac{\tan 4 z}{\tan 7 z} & = & \frac{4 s e c^{2} (0)}{7 s e c^{2} (0)} \\ = & \frac{4}{7} \end{array}$

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