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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 7: Determining limits using algebraic manipulation

# Limits by factoring

In this video, we explore the limit of (x²+x-6)/(x-2) as x approaches 2. By factoring and simplifying the expression, we discover that the function is undefined at x = 2, but its limit from both sides as x approaches 2 is in fact 5.

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Created by Sal Khan.

## Want to join the conversation?

• Does l'Hopital's rule apply here? •  L'Hopitals rule is applicable here

L= lim x->2 for x^2+x-6/(x-2)
L= lim x->2 for f(x)/g(x) where f(x)=x^2+x-6, g(x)=x-2
since lim x->2 f(x)=0 and lim x->2 g(x)=0
and 0/0 is one of the inderminant forms we can apply L'Hopitals rule
f'(x)=2x+1
g'(x)=1
L= lim x->2 for f'(x)/g'(x)=5/1=5
we obtained the same answer when we used factoring to solve the limit

In my opinion, it is easier to use L'Hopitals here than factoring (many will disagree with me).

However, you typically need to know limits before you learn calculus, and you need to know
calculus before you use L'Hopitals rule, so the 1st time people learn how to solve these
types of limits they will use factoring.
• I'm still confuse, is it possible that calculus and algebra can be the same? Some problems i guess can be done through algebra like simple substitution of x=any number. •   Well, in the end, calculus is ultimately based on algebra. It's simply taking a few very useful algebraic functions, making a couple of modifications (like the addition of limits), and expanding on them until they've effectively become their own field of study, in the same way that algebra is ultimately based on arithmetic, but with the added concept of variables. You'd use arithmetic to solve an algebra problem, and you'd use algebra to solve a calculus problem (although, hopefully, you wouldn't use calculus to solve arithmetic). One of the most beautiful parts of mathematics is that it's all connected like that, with one step leading right to the next. So, while they are technically different areas of mathematics, in essence, yes, they are both a part the same thing. One is just a step above the other.
• How did Sal come up with a slope of 1? •   Consider y=mx+c where m is the slope of the line
Now see that when y=x+3 the value of m is 1 which means that slope is 1.
• One question, if you use algebraic rules on it, like sal did and you cancel the (x-2) out, after that the function f(x) is not the same as before or? Because then you dont have that gap anymore? •  Excellent question! And you're right, it's not exactly the same function, because the domain is different -- it now includes x=2, whereas the original function didn't. We can say that it's an extension of the original function, though, because it behaves exactly the same at all the points which were included in the original function.
• Say I have lim of 5/x-7 as x -> 7 I know it is undefined but say I have lim 5/(x-7)^2 as x -> 7 why is it +infinity rather than undefined? • When you're stuck like this, you need to see the limit when approaching to 7 for the right side and for the left. At the first example, lim x->7 (5/(x-7)), if you approach by the left you got -infinite, because you will have a number really close to 7, but still minor than it. And you approach by the right, you got + infinite, because you will have a number really close to 7, but a little higher than it. That's why it's undefined, because the two limits are different. If you do the same thing on the other one, you will get +infinite in the both sides, and the limit is defined as +infinite.
• I didn't get it...when we have f(x) = (x^2 + x -6)/x-2..then our limf(x)as x approaches 2 is not defined..but when w simplified the unction to x+3 then we got a different value of limit as x approaches 2 ..can we have two values for same limit..of the same function? • if
lim f(x) = lim g(x)
x -> 1 x -> 1

can you say that
lim f(x)/g(x) = 1
when x -> 1
is always true?

or it would be correct ONLY as long as lim g(x) does not equal to 0 when x -> 1?

thanks • If we factor x^2+x-6, f(x)=x+3. And why f(x) is still undefined at x=2?   