Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ...

Question

Evaluate the limits in Exercise 7 and 8.

lim (x → ∞) ∫₋ˣ^ˣ sin t dt

Accepted Answer

Welcome back everyone to another video. Compute the limit, limit as x approaches infinity of the integral from negative x up to x of sin of 2 t dt AS 0 B 1 C2 and D does not exist. For this problem we can begin by analyzing the integral. Let's suppose that our integral is equal to i. It is integral from g x up to x. And we're integrating sine of 2 TDT. Notice that the limits of integration are opposite numbers. So whenever this is the case, we want to check whether or not the function that we're integrating is odd. So in this case we're going to take sin of 2 multiplied by negative t. We are replacing t with negative t, which gives us sin of -2t. And remember that sin is an odd function, so we can factor out the negative sign. This is equal to negative sin of tt, right, which is negative of the original function, meaning this function is also odd. And now for odd functions, the integral from negative a up to a of fx dx, where fx is an odd function, this value is zero. The net area is zero. So we can say that the integral in this question is equal to 0 and therefore we are evaluating the limit. As x approaches infinity of zero, now 0 is a constant that is independent of x. So the answer is that constant itself that is equal to 0, and the correct answer corresponds to the answer of choice a. Thank you for watching.

Evaluate the limits in Exercise 7 and 8.lim (x → ∞) ∫₋ˣ^ˣ sin t dt

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Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ^ˣ sin t dt