Determine the following limits.

Assume the f...

Question

Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

Accepted Answer

Welcome back, everyone. Given that our function H satisfies 0 less than R equal to J X less than or equal to 1 minus + X for all X near zero, determine the limit of G of X as X approaches 0. A says it's infinity, B negative infinity, C 0, and the D 1. Now, when we look at our function H here, the truth is we don't really know much about GFX just by looking. However, we, we, we could tell or we could try to figure out the limits for its boundaries. So do we know anything about the boundaries that can help us to find the limit for JF X? Well, to help us, we can use the squeeze theorem. OK. We can use the squeeze theorem. Now, do you remember what that theorem says? Recall that in the squeeze theorem, it states that if F of X, OK, is less than or equal to G of X, which is less than R equal to H of X, OK? In other words, G of X is squeezed between F of X and H of X for all X in some interval around A, except possibly at A itself, OK? And the limit. As X approaches A of F of X is the same as the limit as X approaches A of G of X sorry H X. Which is some value L, OK. Then if both those limits are the same, then the limit as X approaches A for G of X is also going to be equal to L. And now this is a key part of the squeeze theorem here. So what it's basically telling us is that if we can find the limits for F of X and H of X as X approaches 0 as stated in this problem, OK. And if they are the same, then it's going to be the limit for the function in the middle. So now, let's let, OK, 0 be FFX, OK. So let FFX be 0, OK. Or a function that's being squeezed G of X equals G of X, OK. And the function on the upper boundary H X equals 1 minus + X. Let's evaluate the limits for F of X and H of X. No As X approaches 0, let me, let me write that here. As X approaches 0, OK. Then the limit. As X approaches 0 of F of X is going to be the limit as X approaches 0 of 0, and since 0 is a constant, we know this is going to be equal to 0. On the other hand, our limit as X approaches 0 of H of X is going to be the limit as X approaches 0 of 1 minus + X, OK? And no. In this case, if we substitute 0 into our function, that's really 1 minus the cosine of 0, the cosine of 0 equals 1, and 1 minus 1 also equals 0. So now, what does this tell us? Since the limit, as X approaches 0 of F of X is the same as the limit as X approaches 0 of H X, which equals 0, since both those limits are the same, that must mean the limit as X approaches 0 of our function in the middle J of X is also going to be equal to 0. Therefore, the limit is 0. If we look back at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

Key Concepts

Limit of a Function

Squeeze Theorem

Behavior of Sinusoidal Functions

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