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Assume $\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,$ and $\lim _{x \rightarrow 1} h(x)=2 .$ Compute the following limits and state the limit laws used to justify your computations.$$\lim _{x \rightarrow 1} \frac{f(x)}{g(x)-h(x)}$$

8; Quotient and Difference Laws

Calculus 1 / AB

Chapter 2

Limits

Section 3

Techniques for Computing Limits

Campbell University

Harvey Mudd College

University of Nottingham

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we are asked to find the limit as X approaches. One of effort backs over the quantity g of X minus H of X and were also given the values for the individual minutes over here. So using this information, how are we able to solve for this, uh, much larger expression? Well, the first thing that we need to do is we need to employ our limit law of Division, and I'll give you a quick run down and where is limited as X approaches A of FX over some G of X will equal the, uh, the division between the two individual of limits. What I mean by that is the limit of f of X over the limit as X approaches a off G of x. Great. So, knowing this information, let's apply it to this problem right here. Well, first, pretty much I couldn't do it verbatim. Hold on. Let me just box this really quickly. Make sure you guys have this highlighter in your notes. Super, super important. All right, so let's let's let's apply with long division to this one. So what happens is will take the limit of both top and bottom, so it becomes the limit as X approaches one off FX. Well, divine that by the limit as X approaches one of G attacks Linus H of X. Great. Let's first all from the bottom because it seems a little bit more difficult. We already know what the top is. We're just gonna have to play an eight later so we can hold off on doing that. Let's first all from the bottom. So all right, the bottom two side here, let's say limits. As acts approaches one of G of X minus h of X on a solve. This expression will need to employ another one of our limit losses live in los subtraction. Remind you what that is. All write it down for you. Limit as X approaches A of, uh oh, wow. We could use gox, but I'd rather use f of X cause I like Catholics and looks nicer. If that's minus g of X, we can, uh, broke this apart into the difference of the individual of this meaning X minus a X. There's our X to a FX minus that you limit X X approaches a g of X, applying this to original expression that we want to solve for Get the MIT expert. Just one dnx minus the limit as X approaches one, uh, h of X. Great. We actually know what these two values are from Are given. We know that G of X is going to be three and h of x gonna be you, the two. So this simply becomes three. Minus two staples. One Great. So we know that the denominator is going to equal one here. We also know the numerator is going to be equal to eight. So let's plug all this information in will often get eight over. That's a big divisions and look over one. She pulls eight great saw. They're old.

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