82. For what values of a and b is
lim(x→0)(tan(2x/x³) + ...

Question

82. For what values of a and b is

lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?

Accepted Answer

Welcome back everyone. Find the value of the constant a plus b such that limit as x approaches zero of sin 4 x divided by x plus a plus tangent bx divided by x is equal to 2. A says -2, B says 0, C says 2, and D4. So for this problem, let's go ahead and apply the properties of limits. We're going to split this limit into three parts. Limit as x approaches 0 of sin for x divided by x. Plus limit as x approaches zero of a. Plus Limit as x approaches zero of tangent bx divided by x. And now let's continue the equation. We're going to manipulate each limit so that we get appropriate values. For sin 4 x divided by 4 x, what we can do is simply multiply. The numerator by 4 and divide, I'm sorry, and multiply the denominator by 4 as well. The reason why this is useful is because we get sin for x divided by 4 x, right? This is a known limit. And if x approaches 0, then 4 x will approach 0 as well. Let's remember that limit is X approaches 0 of sin x divided by x is equal to 1. So in this case, the main difference is that we have for x. In the numerator for sin for x in the denominator and for x approaches 0, so our limit is equal to 1 and notice that it's basically multiplied by 4. We can always factor out the constant in front of the limit. So we can now show that the first limit is equal to 4 multiplied by 1, right? Because we have a limit in the form of sin x divided by X. In this case it's just 4 x instead of x, right, but it's basically the same angle. So we get 4 multiplied by 1 for the first limit. For the second limit we have a limit of a constant as x approaches 0. It is equal to that constant itself. So that would be a. And now for the third limit, similarly to sin, limit as x approaches 0 of tangent x divided by x is also 1. So we're going to use the same trick. We can multiply the numerator by b and the nominator by b as well, and then we can factor out b, right? We are going to write b in front of the limit. So now tangent bx divided by bx as bx approaches 0 is going to be 1. If x approaches 0, b multiplied by x will also approach 0. So this is consistent with our proposed limit. It is equal to b multiplied by 1. Well done, and we know that this sum is equal to 2. So now let's go ahead and simplify. We got 4 + A + B is equal to 2, and we want to identify A plus B so we can subtract 4 from both sides. 2 minus 4 gives us -2, meaning the correct answer is option A. Thank you for watching.

82. For what values of a and b islim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?

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82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?