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+1 if x≤−1

3 if x>−1; a=−1

Accepted Answer

Welcome back, everyone. We want to analyze the P function HX given by H X equals X minus 2 if X is less than or equal to 0 and 6 if X is greater than 0, where A equals 0. 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. The limit of H of X as X approaches a from the right, and the limit of H X as X approaches A or state that they do not exist. For our answer choices, each of them, each of them shows the possible graph where in A it says that H of A is 6, the limit as it approaches from the left is -2, that as it approaches from the right is 6, and the limit as X approaches a equals -2. B says that they are 226 and 6 respectively, and here is our graph for answer choice B. C says that they are -2, 26, and negative 2. And the D says that they are -2, 26 and that the limit of H X as X approaches A does not exist. And again, here is the graph for answer wd. No, or are we going to analyze our function and then determine. The different values with the help of the sketch of the graph of H. Well, let's make some space down here to do that, OK, and as we said. We know that HFX. Is equal to. X2 minus 2, if X is less than or equal to 0 and 6 if X is greater than 0, where A equals 0. So first, let's do a sketch of our graph and then start to think about the limits and our possible values for our function, OK? Now here we just draw our X and Y axis, OK. Right. Then Label our axis here x and label this one Y. What is our piece wise function telling us? Well, for starters, for X is less than or equal to 0, then it takes on the function X2 minus 2. This is a parabola opening upwards with its vertex shifted 2 units down from the origin. OK? In other words, Here, 2 units down from the origin. We're going to have a parabola that stops here, OK, where? It's going to be X squared. So, for this function, it's X2 minus 2. And it also includes the point where X equals 0 because it's X2 minus 2 if X is less than R equal to 0, OK? Now for X is greater than 0, then our function H of X equals 6. So this is going to be a horizontal line at Y equals 6, but it will not include the value at 06. OK. So here, if this is our 0.06 let me put that in red actually, just to differentiate them, where X is greater than 0, OK. Then we make a space here. Where X equals 6. We're gonna have our spaces and then we're gonna have our line, OK? And this, uh this is 06 here. And this, OK, is going to be H of X where X is greater than 0. Now, now that we have our sketch, we, we have an idea of what our graph looks like. What can we tell? Well, first, notice that the value of the function when X equals A is -2. OK. So, in other words, this tells us that H of A equals -2 at this point on our graph. Next, we also know, OK, that the limit. Of H of X as X approaches A from the right here, sorry, from the left, OK. Is that as it approaches from the left, then it's going to be equal to -2. If we look at it from the right, then the limit as X approaches A from the right of H of X is going to be our function where Y equals 6 or H X equals 6 and approaching from the right, notice that H of X approaches 6, OK? So that limit is going to be equal to 6. And now, since the left hand limit. is not equal to the right-hand limit, then that means the limit, OK, of the function H of X, when X approaches A, in this case, 0, it does not exist, OK? Therefore, from our graph, we know it's going to look something like this where H of A equals -2, the limit of H of X as X approaches A from the left is negative 2. The limit of H of X as X approaches A from the right is 6, and the limit of H X as X approaches A does not exist. Therefore, that tells us that answer choice D. Is the correct answer. And if we look back at the. We can tell that it has this graph, OK. It has our graph similar to what we drew and we can draw the same conclusions. 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+1 if x≤−1
3 if x>−1; a=−1

Key Concepts

Piecewise Functions

Limits

Continuity

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