Determine whether the following functions are continuous at a....

Question

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.

f(x)=2x^2+3x+1 / x^2+5x; a=−5

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says to check the continuity of the function H of X is equal to the quantity of X cubed minus 3 X2 + 2 in quantity divided by the quantity of X2 minus 4 at A equal to 2. Apply the continuity checklist for your analysis. We're given two possible choices as our answers. For choice A, the function is continuous, and choice B, no, the function is not continuous. So the question asks us to determine whether our function is continuous, and here we're looking at the point of X equal to A, where A is equal to 2, and we need to apply our continuity checklist for this point. So, we call for your checklist, we actually have 3 conditions. The first condition that we need to have is that F of A. is defined. The function must be defined at the point. The second condition is that the limit as X approaches A of F of X. Exists. So the limit has to exist, and the third condition is that the limit As X approaches A of F of X. Has to equal F of A. So the limit as X approaches A of F of X has to be equal to the value of the function at that point. So, what we're going to do is take our function and look at each of these conditions, and see if it meets each of the conditions. So we're gonna start off with the first one, which is that our function is defined at a point. So, here we're looking for F of A is being defined. So, A here in this case is going to be equal to. So, we're gonna be looking at the value of H of 2. And so we can use the function that we are given in the problem to determine its value. So, H2 2 is going to be equal to, and here I'll just set X equal to 2 in our function. So that's going to be 2 cubed. Minus 3 multiplied by 2 squad. Plus 2, and that quantity is going to be divided by the quantity of 22 minus 4. Now, if I look at the denominator there, 22 minus 4 is going to be equal to 0. So here I'm actually dividing by 0, and division by 0 is not defined. So that means that our first condition actually fails. So, our function here is actually not continuous. At this point. And we do not need to check the other two conditions, because if any one of the conditions fail, then our function is not continuous. So only if all three conditions are met is our function continuous. So that will save us a little bit of time. And so this is the answer that I'm actually looking for. And this actually corresponds to answer choice B. No, the function is not continuous. So I hope that this explanation has been useful, and I'll see you in the next video.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.
f(x)=2x^2+3x+1 / x^2+5x; a=−5

Key Concepts

Continuity of Functions

Limit of a Function

Rational Functions

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