Suppose g(x) = {2x+1 if x≠0
5 if x=0.

C...

Question

Suppose g(x) = {2x+1 if x≠0

5 if x=0.

Compute g(0) and lim x→0 g(x)

Accepted Answer

Welcome back everyone. Let the function F of X be defined as a piecewise function F of X is equal to X squared plus three, X plus two. If X is not equal to negative two and seven. FF if X is equal to negative two, determine F of negative two. And limit of F of X one X approach is negative two were given four answer choices. They basically define F of negative two. And the limit of F of X one X approaches negative two. For different values, we simply want to identify which one is the correct option. So let's visualize this piece wise function. If we visualize it, we can essentially state that the first part identifies a Parabola X squared plus three, X plus two, right? And this Parabola is not defined that X equals negative two. So let's suppose that we have a point negative two and let's draw a Parabola. So essentially at point negative two, we have a discontinuity. And essentially, if our parabola opens up, we're going to have an undefined point. Let's suppose that it goes down then at the vertex is, it starts to go up and at that point negative two, we are defining a different value of seven, right. So if we approach that point from the left and from the right, we're going to have a limit, right? Because a para is a continuous function, we simply have a point which has an undefined value of F of X. And it's defined at a different value of Y. If we substitute X equals negative gene to the given function, we're not going to get seven. So the limit itself is going to exist because the limit from the left and from the right is going to have the same value. So first of all, let's identify the limit when X approaches negative two of F of X. Since we understand that if we approach negative two from the left and from the right, we're going to get the same value, the limit is going to exist and it is going to be equal to the value of X squared plus three, X plus two at point negative two, we're going to take negative two squared plus three multiplied by negative two plus two. Now let's evaluate the given result four minus six plus two. This gives us zero, even though the value of F at negative two is not going to be equal to zero, right. The limit itself exists and we can evaluate that limit by direct substitution. Now, the value of the function at point negative two is equal to seven. This is given in the piece wise function. And we can clearly see that in our hypothetical graph. So the limit is equal to zero, but the value of the function is equal to seven, they do not necessarily have to be equal to each other and that's perfectly fine. So if we look at the answer choices, we can see that the correct answer to this problem is option A, the value of the function at negative two is equal to seven and the limit of F of X one X approaches negative two is equal to zero. Thank you for watching.

Suppose g(x) = {2x+1 if x≠0
5 if x=0.

Compute g(0) and lim x→0 g(x)

Key Concepts

Piecewise Functions

Limits

Continuity

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