Sketch a graph of f and use it to make a conjecture about the ...

Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

Accepted Answer

Welcome back, everyone. In this problem, we want to analyze the function HX given by H X equals X minus X minus 6 divided by X minus 3, where A equals 3. Sketch a graph of H and determine the values of H of A. The limit of H of X as X approaches A from the left, as X approaches A from the right, and as X approaches A, or indicates they do not exist. For our answer choices, it shows the possible values and our graph, and A says that H of A does not exist. The limit of H of X as X approaches A from the left is 5. Approaches A from the right is 5, and as it approaches A is 5. B says that they are 475, and 6 respectively, and here is the graph. C says they are 586, and 9, and here is the graph. And then D says that H of A does not exist while the limits are 55 and 5 respectively, and here is our graph. Now below these I've also put a graph, OK, that we're going to use to help us to sketch H of X. So let's go ahead and try to sketch the graph and then start to think about the values of the function and the limits as needed. So we know that so far. We were given 8 of X. Being equal to X2 minus X minus 6 divided by X minus 3, OK, where A is equal to 3. Now for this function. Notice here that by quadratic factorization, we could rewrite the numerator as X minus 3 multiplied by X + 2 divided by X minus 3. And since X minus 3 is a common factor in the numerator and denominator, we can cancel to get H of X to be equal to X + 2. So what this tells us then is that our graph is really a straight line in the form X + 2. However, because we canceled X minus 3 from our function, then there's going to be a hole at the point where X equals 3, OK? And at X equals 3, Age of 3 would be equal to 3 + 2, which is 5. Therefore, on our graph, 35 is going to be a whole. Now how could you draw the line Y equals X + 2? Well, given that it's X + 2, we already have one coordinate. We know that the Y intercept is 2, so 02 is on the graph. Now, we could arbitrarily choose some other value. So let's say that we choose X to be -2, and if X is -2, 2 + 2 equals 0. So our Y value is going to be 0. Thus, 02 and negative 20 are on the graph, OK? So now that we have both of those values, we can plot those points. So 02, OK, and the negative 20. I know we can connect those points, OK, to plot over line Y equals X + 2. OK. So it's gonna look like this and this direction. And now remember we said that there's going to be a hole. So I should probably write that. At 35, OK. So, let's go to 3 500 graph. Let's make some space here and let's just draw our hole, OK? So it's gonna look something like that. No, Since we have our graph, we can start to figure out the values we are interested in, that is H of A. In this case, we know that A is 3, so as a matter of fact, let me write that here since A equals 3. That means we're really looking for age of 3, OK. We're looking for the limit of H of X as X approaches 3 from the left. The limit of 8 X. As X approaches 3 from the right. And then the limit of H of X. As X approaches 3, OK? So we're going to get all of these values by looking on our graph. Now let's start with age of 3. When X equals 3, we know that there's a hole in our graph, OK? So that means A + 3 is undefined. It does not exist, OK? Next, thinking about the limit of H of X as X approaches 3 from the left. As X approaches 3 from the left, notice that H of X approaches 5. Thus, the limit is equal to 5. As H of X approaches 3 from the right, OK. Not that, sorry, as X approaches 3 from the right, notice H of X also approaches 5. Thus, the limit is 5. And not to find the limit of a traffic as X approaches 3. Then, since our left-hand limit equals our right hand limit, which are both equal to 5, then our limit as X approaches 3 is also equal to 5. Thus, H of 3 does not exist. The limit of H X as X approaches 3 from the left as it approaches 3 from the right, and as it approaches 3 are all equal to 5. If we go back to our answer choices, OK, and look at our graphs and our results. Then we can tell that answer choice A, OK, answer choice A is the correct answer. Thanks for watching everyone. I hope this video helped.

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = x^2+x−2 / x−1; a=1

Key Concepts

Limits

Continuity

Graphing Rational Functions

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