Tired of ads?
Join today and never see them again.
Time to crank this these limits up a notch or two. Or five. Five notches of cranked-upedness, one for each of the five properties of limits. Let's roll.
Check it out: a wild limit appears.
We'll use the Constant Multiple Rule on this limit. The idea is that we can "pull a constant multiple out" of any limit and still be able to find the solution.
Now that we've found our constant multiplier, we can evaluate the limit and multiply it by our constant:
You might be thinking we're as crazy as the day is long, because all we have to really do in this problem is use direct substitution. But what about when we don't actually know the function?
If , what is ?
We couldn't solve this before, but now it's trivial. It can't even make us sweat. What does make us tense up is the Constant Multiple Rule written in formula-speak:
If b and c are constants and then .
See, doesn't it make your eyes water just looking at it?
How about another limit property? That seems appropriate, for some reason.
Yes, we know that can use direct substitution here, too. However, you can also use the Sum Rule limit property. It says that:
If and then .
In normal human terms, this property basically means that any time we have two functions added together, we can just evaluate each limit and then add the two solutions together. Let's go back to our problem and apply our newfound knowledge.
Boom. We take one limit, and we split it into two smaller, easier to work with ones.
We like smaller, easier to work with problems.
By now you may have guessed that we're now going to apply the Product Rule for limits. Nice guess; what gave it away? It says:
If and then .
Just like the Sum Rule, we can split multiplication up into multiple limits. Just be careful for split ends.
Before we move on to the next limit property, we need a time out for laughing babies. Because if there is one thing better than laughing limits, it's laughing babies.
The Power Rule is the Product Rule on steroids. Just don't let the IOC catch you using it.
If then .
Power rule time. This property is also very, very straightforward.
Just lift the exponent up out of the limit. Remember to bend at the knees, not at the waist.
Division presents a special problem, because if we end up with a limit that equals zero in the denominator of a fraction, the world more or less comes to an end. That sounds like a bad idea to us.
Enough with the pleasantries, here is the Quotient Rule:
If and then given K ≠ 0.
Now that we have the unpleasantries out of the way, we can show you what we mean.
When we have a fraction (i.e., division) within a limit, we can instead find the limits of the top and the bottom on their own.
And we have our answer. We were lucky that the zero was on top of the fraction, and not on the bottom. Soon, though, very soon, we'll be able to find the limits even in that kind of situation. Our limit-finding power will be limitless.