What's going to happen as x approaches positive infinity? Maybe candy will fall out of the function. Let's start off by plugging in a big number, like 10,000.
That is not candy. That is a really big negative number, and it's only going to get worse as x gets even bigger.
We can also think of this function as a fraction, with a denominator of 1. Then, as long as we realize that has the largest degree ( is larger than 1), we'll find the same result.
Example 2
Evaluate .
We have a limit that goes to infinity, so let's start checking some degrees. It's like we're a bouncer for a fancy, PhD-only party.
The largest degree is 2 for both up top and down below. They are equal. The limit will be the ratio of the leading coefficients. We have 4 over 2, which means that the limit as x approaches infinity is 2.
Example 3
Evaluate .
The degree of the numerator of 1, while the degree of the denominator is 2. They're both 3 degrees away from Kevin Bacon, though.
This function is bottom heavy, so the denominator is becoming obscenely huge while the numerator is only large. That means the function will approach zero as x approaches infinity.
Example 4
Evaluate .
This question is a tough one. How can we possibly deal with sin(x)? The secret is that it is never going to get larger than 1 or smaller than -1. The denominator, though, is going to just keep growing and growing.
The fraction is going to continue to change signs (not sines, but because of it), but it will keep getting smaller and smaller as x gets bigger.
Example 5
Evaluate .
Checking our fraction, we have equal degrees. Checking our limit, we go to negative infinity. Proceed with caution, Shmooper.
If we just look at the coefficients, the ratio is 2. However, we have to check what sign our answer will have. When something is cubed, the sign stays the same. x is becoming really negative, so we have , which hey, is positive. That means:
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has the largest degree (
is larger than 1), we'll find the same result.
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, which hey, is positive. That means: