L'hopsitals rule

**johntuan** · October 30th, 2008, 08:04 PM

1. lim x->0 [sin4x/tan5x]

i have no idea how to do this problem. any help would be greatly apprecited.

**Chop Suey** · October 30th, 2008, 08:10 PM

$\frac{\sin{4x}}{\tan{5x}}$

$= \cos{5x} \cdot \frac{\sin{4x}}{\sin{5x}}$

$= \frac{4}{5} \cos{5x} \cdot \frac{\displaystyle\frac{\sin{4x}}{4x}}{\displayst yle\frac{\sin{5x}}{5x}}$

**Prove It** · October 30th, 2008, 08:39 PM

Originally Posted by johntuan

1. lim x->0 [sin4x/tan5x]

i have no idea how to do this problem. any help would be greatly apprecited.

$\lim_{x \to 0} \frac{\sin{4x}}{\tan{5x}} = \lim_{x \to 0} \frac{4\cos{4x}}{\frac{5}{\cos^2{5x}}}$

$=\lim_{x \to 0} \frac{4}{5} \cos{4x} \cos^2{5x}$

$= \frac{4}{5} \cos{0} \cos^2{0}$

$= \frac{4}{5}$

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