How do you compute the limit of #cot(4x)/csc(3x)# as #x->0#?

Answer 1

#= 3/4#

# lim_(x to 0) cot(4x)/csc(3x)#
#=lim_(x to 0) ( cos(4x) sin(3x))/(sin (4x) #
#=lim_(x to 0) cos(4x) ( 3x(sin(3x))/(3x))/(4x(sin (4x))/(4x)) #
#=lim_(x to 0) cos(4x) ( 3(sin(3x))/(3x))/(4(sin (4x))/(4x)) #

The limit of a product/quotient is the product/quotient of the limits:

#=lim_(x to 0) cos(4x) cdot ( 3lim_(x to 0) (sin(3x))/(3x))/(4lim_(x to 0) (sin (4x))/(4x)) #

Now:

#lim_(z to 0) (sin z)/z = 1#

And:

#=lim_(z to 0) cos(z) = 1#

So we have:

#=1 cdot ( 3 cdot 1)/(4 cdot 1) = 3/4 #
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Answer 2

To compute the limit of cot(4x)/csc(3x) as x approaches 0, we can simplify the expression using trigonometric identities.

Recall that cot(x) is equal to 1/tan(x), and csc(x) is equal to 1/sin(x).

Using these identities, we can rewrite the expression as (1/tan(4x))/(1/sin(3x)).

Next, we can simplify further by multiplying the numerator and denominator by sin(4x)/sin(4x), which gives us sin(3x)/(tan(4x)*sin(4x)).

Now, we can use the fact that sin(x)/x approaches 1 as x approaches 0, and tan(x)/x also approaches 1 as x approaches 0.

Therefore, the limit of cot(4x)/csc(3x) as x approaches 0 is equal to sin(3x)/(tan(4x)*sin(4x)), which simplifies to 3/4.

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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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