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 sin θ cot 2θ

Accepted Answer

Welcome back, everyone. Determine the limit. Limit as X approaches 0 of sin of 2 X multiplied by cotangent of 3 X. We're given 4 answer choices A says 2/3, B1, C 3 halves, and D does not exist. So let's begin solving for this limit with direct substitution, assuming that our function is continuous at X equals 0. This gives us sine of 0, multiplied by cotangent of 0. Now sin of 0 is equal to 0, and cotangent of 0, well, essentially it might be positive or negative infinity. Depending on which side we are using. To approach X equals 0 from right. Essentially we can approach 0 from the left or from the right. So this is an indeterminate form, and what we have to do is basically use another method to evaluate the limit because our function is not continuous at X equals 0. What we have to notice is that X approaches 0, and this means that X is small. Now, if X is small, it basically means that we can approximate our functions. Let's recall that sine of X is approximately equal to X if X is close to 0, and then cosine of X is approximately equal to 1. And we're going to rewrite our limit in terms of signs and cosines. So we have limit as X 0 of sin of 2 X multiplied by cotangent, which is the ratio between cosine and sign. So we get cosine of 3 X divided by sin of 3 X. And now using our approximation, we can say that this limit would be equal to limits x approaches 0 of 2 X. Notice that X approaches 02 X will approach 0 as well, and so will 3X, right? So we get sin of 2 X being approximately equal to 2 X. And we are essentially multiplying that by cosine of 3 X which is going to be approximately equal to 1 divided by sin of 3 X, which is basically approximately equal to 3 X. And then we can clearly see that our axis will be canceled out. And the limit becomes limit as x approaches 0 of 2/3, which is a constant, and we have to recall that the limit of a constant is the constant itself because it's independent of X. So the correct answer to this problem corresponds to the answer choice a 2/3. Thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ cot 2θ

Key Concepts

Limit of a Function

Trigonometric Identities

Squeeze Theorem

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