How do you find the limit of #sin(costheta)/sectheta# as #theta->0#?

Answer 1

Substitution.

As #theta rarr0#, we have #cos theta rarr1# and #sec thetararr1#, and the sine function is continuous. So the limit is #sin(1)/1 = sin1#.
(That is the sine of the number #1# or of an angle of measure #1# radian. Since #1# is somewhat close to #pi/3#, #sin1# is somewhat close to #sqrt3/2#)
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Answer 2

To find the limit of sin(theta)/sec(theta) as theta approaches 0, we can simplify the expression using trigonometric identities.

Recall that sec(theta) is equal to 1/cos(theta).

Substituting this into the expression, we have sin(theta)/(1/cos(theta)).

To simplify further, we can multiply the numerator and denominator by cos(theta), which gives us sin(theta) * cos(theta)/1.

Using the double-angle identity for sine, sin(2theta) = 2sin(theta)cos(theta), we can rewrite the expression as 2sin(theta)cos(theta)/1.

Now, as theta approaches 0, sin(theta) and cos(theta) both approach 0.

Therefore, the limit of sin(theta)/sec(theta) as theta approaches 0 is equal to 2(0)(0)/1, which simplifies to 0.

Hence, the limit is 0.

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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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