Use a graph of f to estimate

Question

\lim_{x \to a} f (x)

or to show that the limit does not exist. Evaluate f(x) near

x = a

to support your conjecture.

$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

Accepted Answer

Hello there today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function F of X is equal to one minus cosine of four, X minus four divided by X minus one to the power of two, use its graph to estimate the limit as X approaches one evaluate F of X for values near X equals one to support your estimate. Awesome. So it appears for this particular prong we're trying to use our function, which we're gonna use the function to create a graph and using our graph, we're gonna estimate the limit because the limit value is what we're ultimately trying to solve for us. We're trying to figure out what the estimate of the limit is provided the conditions set to us by the problem itself. So now that we know that we're trying to figure out what the limit is. Let's read off our multiple choice answers to see what our final answer might be. A is seven, B is six C is eight and D is DNE, where DNE means does not exist. OK. So first off, in order to help us solve for this particular problem, let's take our function and let's plug it into our graphing calculator or whatever graphing software you're familiar and comfortable with. So I've gone ahead and plugged it into a graph, graphing calculator. And the following graph is what I came up with or what my calculator came up with. So as you could see in our graph, we have one tall, one large peak that is at about it looks like one eight. So one comma eight, we have a high peak and then we have little peaks that trend and then go as and then start to trail out and become a straight line. So it goes so it's only like it starts with one big peak and then it like willows out until it flattens out going both directions. So as we can see from our graph, we have the one large peak like we discussed. But it's important to note and recall that to estimate the limit as X approaches one, we can look at the graph of the function of F FX. And as we can see, as X approaches one from the left side F of X approaches eight. So as X approaches one from the right side, F of X approaches eight, so let's make a note of this down here. So as X approaches one, so as X approaches one from the left side. So I'll do a capital L for left, we could see that F of X approaches eight. And as X approaches one from the right side, which I will do a capital R for right F of X will approach eight. Fantastic. So with that said, we can put this into writing and we could say that the limit as X approaches X from the left, sorry as X approaches one from the left side. So the limit as X approaches one from the left of F of X, you could say that this is equal to the limit as X approaches one from the right side of F of X is equal to eight. So therefore, without a doubt, as we should recall, since the left hand limit, so left hand limit is equal to the right hand limit, which is RHL. So because the left hand limit is equal to the right hand limit, the given limit will exist in equal eight. So this will mean that it will exist in equal at eight. So we'll put a nice, we'll put a number eight here and that is our final answer. Hooray, we did it. So in order to help us support our estimate, let's go ahead and refer to this graph right here. So let's say we made a table to help us make sure that our specific answer since it said support our estimate. So in order to get better support our estimate you can put together a little table that has different values for X. And then when you plug in the value for X into your equation of F of X, you will gain the following values. So for X, we, when you plug in a value of X of 0.99 you would get for F of X would be equal to 7.998933 and et cetera. So you could create like a table to help you prove your estimate. And then you just plug in different values for X into your function of F of X. And then that would be your total when you plug in your value of X. That's why with X is a smaller number compared to F of X because you're plugging X into your function and that's it. So looking at your multiple choice answers, the correct answer has to be multiple choice answer letter C which is eight and that's it. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.
$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

Key Concepts

Limit of a Function

Continuity

Graphical Interpretation of Limits

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