Let f(x) = {x^2+1 / if x

Question

Let f(x) = {x^2+1 / if x<−1

√x+1 if x≥−1.

Compute the following limits or state that they do not exist.

limx→−1 f(x)

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with limits. This problem says, consider the function G of X defined as follows G of X is equal to X cubed minus two if X is less than two and the square root of the quantity of X minus two, if X is greater than, or equal to two, determine the value of the limit. Uh As X approaches two of G of X or state that it does not exist. We're given four possible choices as our answers. For choice A zero, for choice B six, for choice C one and for choice D, the limit does not exist. So this question asks us to find the value of the limit as X approaches two of G FX. But we really need to first determine whether this limit exists or not. So we call that a limit to exist that the limit value should be the same whether you approach from the left or right side of the function. So we're actually going to have to do two limits here. One as we approach X equal to two from the left and the other as we approach X equal to two on the right. So we'll start off by approaching um X as it goes towards two from the left. So we're gonna be looking at the limit as X approaches two from the left. And here, since we're approaching from the left, X is going to be less than two. So that means we're going for G of X, we're gonna have the function of X cubed minus two. Now, we can evaluate this limit just by direct substitution. So we'll set X equal to two. And so this is gonna be equal to two cubed minus two, which is gonna be equal to six. So that's gonna be our limit. As we approach to from the left, we do the same thing from the right. So we'll have the limit as X approaches two from the right. And here X is going to be greater than two. And so for G of X, we're going to have the square root of the quantity of X minus two. Again, we can evaluate this limit by direct substitution. So we'll set X equal to two. So we're going to have this equal to the square root of two minus two, which is going to be equal to zero. So those are two values for our limit as we're approaching from the left and from the right. However, these two values are different and in order for a limit to exist, the value must be the same whether I approach the um point either from the left or from the right side of the function. So since these two values are different, that means that the limit does not exist. So the correct answer to this problem is choice D so I hope that this explanation has been useful and I'll see you in the next video.

Let f(x) = {x^2+1 / if x<−1
√x+1 if x≥−1.

Compute the following limits or state that they do not exist.
limx→−1 f(x)

Key Concepts

Piecewise Functions

Limits

Left-Hand and Right-Hand Limits

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