Thread: Trig Derivatives
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Old October 2nd, 2007, 09:34 AM
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Quote:
Originally Posted by coolio View Post
2. lim t -> 0 tan6t / sin 2t
I'll borrow a trick from one of the other members here:
\lim_{t \to 0} \frac{tan(6t)}{sin(2t)}

= \lim_{t \to 0} \frac{\frac{6t}{6t} \cdot tan(6t)}{\frac{2t}{2t} \cdot sin(2t)}

= \lim_{t \to 0} \frac{6t}{2t} \cdot \frac{\frac{tan(6t)}{6t}}{\frac{sin(2t)}{2t}}

= 3 \cdot \lim_{t \to 0} \frac{\frac{tan(6t)}{6t}}{\frac{sin(2t)}{2t}}

Now,
\lim_{t \to 0}\frac{tan(6t)}{6t} = 1
and
\lim_{t \to 0}\frac{sin(2t)}{2t} = 1

So
\lim_{t \to 0} \frac{tan(6t)}{sin(2t)} = 3 \cdot \lim_{t \to 0} \frac{\frac{tan(6t)}{6t}}{\frac{sin(2t)}{2t}} = 3 \cdot \frac{1}{1} = 3

-Dan
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