Limits at Infinity - Online Tutor, Practice Problems & Exam Prep

Understanding limits as x approaches infinity or negative infinity is crucial in calculus. When evaluating limits of rational functions, divide each term by the highest power of x. If the degree of the numerator is less than the denominator, the limit approaches 0. If the numerator's degree is higher, the limit does not exist, tending towards positive or negative infinity. For equal degrees, the limit is the ratio of the leading coefficients. This process aids in analyzing function behavior and ensures clarity in determining limits in various scenarios.

Evaluate the following limit: ${\lim }_{x\to \infty }-\frac{4}{x}+2$.

• A. 0
• B. 2
• C. -2
• D. 6

Evaluate the following limit: $\lim_{x\rarr\infty}\left(x\right)-6$.

• A. 6
• B. -6
• C. 0
• D. Does not exist

Finding Limits at Infinity Algebraically

Welcome back, everyone. So in recent videos, we talked about how to find limits at infinity. And recall when solving these types of problems, we were given either a table or a graph, and we needed to figure out how the function behaved as \( x \) got infinitely big or infinitely small. Now in this video, we'll see situations where we don't have a graph or table to help us, but we simply just have these types of rational functions that look a bit complicated. Now if this sounds scary, don't sweat it, because it turns out this process for solving limits purely algebraically with these types of rational functions is actually quite straightforward, and we're even going to learn a shortcut, which allows us to solve these problems super fast. So without further ado, let's jump right into some examples of this.

Now we'll start over here at example a, where we're given this rational function and that we have the limit as \( x \) is approaching infinity. Now whenever you have to solve these types of limits for these rational functions, what you need to do is take each term and divide it by the highest power of \( x \). So looking at this example right here, I can see that the highest power of \( x \) that we have is this \( x^3 \). So what that means is I need to take every term on top and divide it by \( x^3 \), and I need to take every term on bottom and divide it by \( x^3 \). This is the strategy. So, I'll rewrite this limit down here. We have the limit as \( x \) is approaching infinity, and what we're going to have is this \( \frac{1}{x^3} \) distributed into every single term. So, we'll have \( \frac{3x^2}{x^3} \), which turns out to just be \( \frac{3}{x} \). Then we're going to have \( \frac{4x}{x^3} \), which is \( \frac{4}{x^2} \), and we're going to have \( -\frac{1}{x^3} \), and then this whole thing is going to be divided by \( \frac{x^3}{x^3} \), which is 1, plus \( \frac{27}{x^3} \). Now recall that whenever you have \( x \)'s on the bottom of the fraction, and you don't have any \( x \)'s on top, if \( x \) is approaching infinity, then the whole thing is just going to go to 0 because the bottom is infinitely growing where the top is not. So in this situation, what we can see here is that this is all just gonna go to 0. That's gonna go to 0, that's gonna go to 0, and that's gonna go to 0. So all we're going to end up with is 0 on top and just one on the bottom, and \( \frac{0}{1} \) is 0. So that right there is the solution to this limit. So this is how you can solve these types of problems where you have \( x \) approaching infinity and you have these complicated rational functions.

Now you may have noticed that this process was a bit on the te

Evaluate the following limit: $\lim_{x\rarr\infty}\frac{x+7}{8x-100}$.

• A. $7$
• B. $-\frac{1}{100}$
• C. $-\frac{7}{100}$
• D. $\frac{1}{8}$

Finding Limits at Infinity Algebraically Example 1

Welcome back, everyone. So, in this example here, we have these three situations where we're asked to find the limit of our function, and for all of these, we have x either approaching negative infinity or infinity. Now, what we can do is use our shortcuts for solving rational functions where you have limits that go to infinity. And recall that for these shortcuts, it doesn't matter whether we have negative infinity or positive infinity, so we just need to look for whatever the highest x power is and figure out how to solve it from there. So let's get started. Now we'll start with example a. Example a, we need to find the limit as x approaches negative infinity for this entire function right here. Now to solve this, I'm looking for the x that has the highest power. I can see the highest power x we have on top is x squared. The highest power x we have on the bottom is also x squared. So we have equal exponents on top and bottom. We have the

e , so what that means is that the solution is going to be a ratio between the coefficients out in front, which in this case is 7 49

And this actually reduces, because 49 is the same thing as 7 times 7, so we can cancel one of the sevens here, meaning our solution is really

1 7

And that is the solution right there to example a. Now let's move on to example b. Example b is going to be this situation right here, where we have 5x squared plus 3. Now this might not really look like a rational function per se, but if you wanted to, you could put this whole thing over 1 to turn it into a rational function. Now if we do that, notice we have this 5x squared on top, and then that's the highest exponent we have is x squared. Since we have an x squared on top, and we don't have any x's on the bottom, obviously we just have 1. Then you can imagine here that this x squared is, we have the x approaching negative infinity, and then squaring it is just going to give you a bigger and bigger number. That means the entire thing's going to blow up to infinity, meaning that our limit here does not exist. And that was actually the rule. Remember that if you have the highest exponent show up on top, then the entire thing is going to blow up to an infinite or negative infinite value, which does not exist. So that's our solution for these types of situations. But now let's take a look at example c. Example c gives us this giant situation right here, this giant rational function, and we have x approaching positive infinity. Now what I need to do is look for my highest exponent. Now you can see the highest exponent that we have on top is this x to the 10th power. I can see the highest exponent we have on the bottom, well, that's also x to the 10th power. So because we have equal exponents on top and bottom, we need to find a ratio of their coefficients, which is going to be

10 23

And that right there is the solution to example c, and that is how you can solve these types of problems. So as you can see, it's very quick to do these problems when you use this shortcut. You just find wherever the highest x is, and then you remember your rules. If it's on bottom, it goes to 0. If it's on top, it goes to infinity. And if it's equal, then you get a ratio between the coefficients out in front. So, I hope you found this video helpful. Thanks for watching, and let's move on.

Here’s what students ask on this topic:

What is the limit of 1/x as x approaches infinity?

To find the limit of $\frac{1}{x}$ as $x$ approaches infinity, consider what happens to the fraction as $x$ gets larger. As $x$ increases, the denominator grows, making the fraction smaller. Mathematically, $lim\left(\frac{1}{x}\right)=0$. Therefore, the limit of $\frac{1}{x}$ as $x$ approaches infinity is 0.

How do you find the limit of a rational function as x approaches infinity?

To find the limit of a rational function as $x$ approaches infinity, divide each term by the highest power of $x$ in the denominator. For example, for $\frac{3x^2+4x-1}{x^3+27}$, divide each term by $x^3$. Simplifying, you get $\frac{\frac{3}{x}+\frac{4}{x^2}-\frac{1}{x^3}}{1+\frac{27}{x^3}}$. As $x$ approaches infinity, terms with $x$ in the denominator approach 0, so the limit is 0.

What happens to the limit of a function if the degree of the numerator is higher than the degree of the denominator?

If the degree of the numerator is higher than the degree of the denominator in a rational function, the limit as $x$ approaches infinity does not exist. This is because the numerator grows faster than the denominator, causing the function to tend towards positive or negative infinity. For example, for $\frac{x^4}{x^2}$, the numerator's degree (4) is higher than the denominator's degree (2), so the limit as $x$ approaches infinity is infinity, meaning the limit does not exist.

How do you find the limit of a function as x approaches negative infinity?

To find the limit of a function as $x$ approaches negative infinity, analyze the function's behavior as $x$ becomes very large in the negative direction. For rational functions, divide each term by the highest power of $x$ in the denominator. For example, for $\frac{3x^2+4x-1}{x^3+27}$, divide each term by $x^3$. Simplifying, you get $\frac{\frac{3}{x}+\frac{4}{x^2}-\frac{1}{x^3}}{1+\frac{27}{x^3}}$. As $x$ approaches negative infinity, terms with $x$ in the denominator approach 0, so the limit is 0.

What is the shortcut for finding limits of rational functions as x approaches infinity?

The shortcut for finding limits of rational functions as $x$ approaches infinity involves comparing the degrees of the numerator and the denominator. If the degree of the numerator is less than the denominator, the limit is 0. If the degree of the numerator is higher, the limit does not exist. If the degrees are equal, the limit is the ratio of the leading coefficients. For example, for $\frac{2x+1}{5x-1}$, the degrees are equal (1), so the limit is $\frac{2}{5}$.