Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.4 - Indeterminate Forms and l''Hospital''s Rule - 4.4 Exercises - Page 312: 65

Answer

$$y=e^4$$

Work Step by Step

$\lim\limits_{x \to 0^+}$ $(4x+1)^{cotx}$= $$Let y= (4x+1)^{cotx}$$ By taking $lny$ and $ln(4x+1)^{cotx}$ and taking $\lim\limits_{x \to 0^+}$ to both $$\lim\limits_{x \to 0^+}= \lim\limits_{x \to 0^+} cotx ln(4x+1)= 0*\infty$$ $$lny= \lim\limits_{x \to 0^+} \frac{ln(4x+1)}{tanx}$$ by using L'Hopital's Rule $$lny= \lim\limits_{x \to 0^+} \frac{4}{\frac{4x+1}{sec^2x}}$$ $$=\lim\limits_{x \to 0^+} \frac{4sec^2x}{4x+1}$$ $$lny= \frac{4(1)}{4(0)+1}=4$$ $$y=e^4$$
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