Using limθ→0 sin θ / θ = 1

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Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 tan θ / θ²cot 3θ

Accepted Answer

Welcome back, everyone. Evaluate the limit below. In this problem, we're given the limit as x approaches 0 of tangent of 5 X divided by X quad cotangent of 7 X, and we're going to begin with a re substitution, assuming that our function is continuous at X equals 0. This gives us tangent of 0, which is 0, divided by 02, which is. multiplied by cotangent of 0, which can be either positive or negative infinity, depending on how we approach X from the left or from the right. In any case, this is an indeterminate form and we have to notice that X approaches 0. So X is small, right? It is close to 0. And therefore, if X is small, we have to recall that sin of x is approximately equal to x. And what about cosine? Well, it is approximately equal to 1 if X is small, and we're going to use those approximations to rewrite our limit. So we get limit as X approaches 0. 1st of all, we're expressing tangent in terms of the ratio between the sin of 5 X and cosine of 5 X. And now considering the denominator, we have X2d multiplied by cotangent, which is the ratio between. Cosine of 7 X and sine of 7 X. Moving on, we're going to use the approximations. This gives us limit as X approaches 0. It is important to understand that 5 X and 7X will also approach 0 if X approaches 0. So, sine of 5 X is going to be approximately equal to 5 X. We're dividing by cosine of 5 X which is going to be approximately equal to 1. And then dividing by X2 multiplied by cosine of 7 X, which is approximately equal to 1, divided by sin of 7 X which is approximately equal to 7 X. And now we can simplify. Limit SX approaches 0 of 5 X divided by X2 divided by X gives us X, divided by 7. Now we're using the reciprocal rule, while we can essentially divide, right? 5 X divided by X divided by 7, is basically Limit as X approaches 0 of 35. X Divided by X and we can clearly cancel out the x term. Which gives us the limit of a constant, which is that constant, right? Because it's independent of X. So the limit is 35. Thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 tan θ / θ²cot 3θ

Key Concepts

Limit of a Function

Trigonometric Limits

L'Hôpital's Rule

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