How do you evaluate the limit #tan(4x)/x# as x approaches #0#?

Answer 1

The limit equals #4#.

Rewrite in sine and cosine using the identity #tanx = sinx/cosx#.
#=lim_(x -> 0)(sin(4x)/cos(4x))/x#
#=lim_(x->0) sin(4x)/(xcos(4x))#
Rewrite so that that one expression is #sin(4x)/x#.
#=lim_(x-> 0) sin(4x)/x xx 1/cos(4x)#
Use the well know limit that #lim_(x ->0) sinx/x = 1# to deduce the fact that #lim_(x -> 0) sin(4x)/x = 4#.
#=4 xx 1/cos(0)#
#=4 xx 1#
#= 4#

Hopefully this helps!

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Answer 2

To evaluate the limit of tan(4x)/x as x approaches 0, we can use L'Hôpital's Rule. Taking the derivative of both the numerator and denominator, we get 4sec^2(4x)/1. Substituting x=0 into this expression, we find that the limit is 4. Therefore, the limit of tan(4x)/x as x approaches 0 is 4.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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