John J.

asked • 07/20/17

Calculus Question

Evaluate the following limit:
 
lim of sin(4theta)/(5theta) as theta approaches 0.
 
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Inactive Tutor answered • 07/20/17

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Since this is an indeterminate form, you can use L'Hopital's rule.  Are you familiar with it?
 
lim θ→0 of (sin4θ / 5θ) = lim θ→0 of (4cos4θ / 5) = 4/5
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If you do not know L'Hopital's rule, you can use
 
lim x→0 of (sinx / x) = 1
 
Then 
 
lim θ→0 of (sin4θ / 5θ) = lim θ→0 of [(4/4)(sin4θ / 5θ)] = lim θ→0 of [(4/5)(sin4θ / 4θ)] 
 
= (4/5)[lim θ→0 of (sin4θ / 4θ)] = (4/5)(1) = 4/5
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07/20/17

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