Finding the limit of a rational expression

In summary: So in your function, if x > 0, then the denominator is positive, but the numerator is positive if x > -1 and negative if x < -1. This means that as x → ∞, the function approaches 1 from the positive side. Similarly, as x → -∞, the function approaches -1 from the negative side. Therefore, the horizontal asymptotes are y = 1 and y = -1.In summary, to find the horizontal asymptotes of a rational function, we can take the limit as x goes to infinity and multiply the numerator and denominator by ##\frac{1}{x}##. However, we must be careful because ##\sqrt{x^2}## is equal to ##
  • #1
Mr Davis 97
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To find the horizontal asymptotes of a rational function, we find the limit as x goes to infinity. Given the rational function ##\displaystyle\frac{x + 1}{\sqrt{x^2+1}}##, we can find the limit by multiplying the numerator and the denominator by ##\frac{1}{x}##. This gives us ##\frac{1 + \frac{1}{x}}{\sqrt{1+\frac{1}{x^2}}}##. Taking the limit of this gives us 1, so it would seem as though the horizontal asymptote is 1. However, looking at the original function, it is obvious that if we went in the negative direction to infinity the number would be negative. Therefore, what I am doing wrong? Why doesn't the process of dividing the numerator and denominator yield the correct answer?
 
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  • #2
Mr Davis 97 said:
To find the horizontal asymptotes of a rational function, we find the limit as x goes to infinity. Given the rational function ##\displaystyle\frac{x + 1}{\sqrt{x^2+1}}##, we can find the limit by multiplying the numerator and the denominator by ##\frac{1}{x}##. This gives us ##\frac{1 + \frac{1}{x}}{\sqrt{1+\frac{1}{x^2}}}##. Taking the limit of this gives us 1, so it would seem as though the horizontal asymptote is 1. However, looking at the original function, it is obvious that if we went in the negative direction to infinity the number would be negative. Therefore, what I am doing wrong? Why doesn't the process of dividing the numerator and denominator yield the correct answer?

You can only divide whatever's under the radical sign by ##x^2## if you're sure that ##\frac{1}{x}## is positive, which isn't always the case. Hint: ##\sqrt{\frac{1}{x^2}}## is, in general, equal to ##|\frac{1}{x}|##, NOT ##\frac{1}{x}##.
 
  • #3
In your rational function, the denominator is always positive, but the numerator is positive if x > -1, and is negative if x < -1. There are two horizontal asymptotes: y = 1 (as x → ∞) and y = -1 (as x → -∞).

If you factor the numerator and denominator, you get ##\frac{x(1 + 1/x)}{|x|\sqrt{1 + 1/x^2}}##.
##\frac x {|x|} = 1## if x > 0, but ##\frac x {|x|} = -1## if x < 0.

Note that ##\sqrt{x^2} = |x|##, not x.
 
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Related to Finding the limit of a rational expression

What is a rational expression?

A rational expression is a fraction with polynomials (expressions with variables) in the numerator and denominator. It can also be written as a ratio of two polynomial expressions.

What is the limit of a rational expression?

The limit of a rational expression is the value that the expression approaches as the independent variable (usually denoted as x) gets closer and closer to a certain value. This is also known as the limit of a function.

How do you find the limit of a rational expression?

To find the limit of a rational expression, you can first try to evaluate the expression at the given value of the independent variable. If this results in a defined value, then that is the limit. If not, you can use algebraic methods such as factoring and canceling common factors to simplify the expression and find the limit.

When does the limit of a rational expression not exist?

The limit of a rational expression does not exist when the expression has a vertical asymptote at the given value of the independent variable. This means that the value of the expression approaches positive or negative infinity as the independent variable approaches the given value.

Why is finding the limit of a rational expression important?

Finding the limit of a rational expression is important in calculus and many other areas of mathematics. It allows us to understand the behavior of a function at a specific point and to make predictions about the function's values near that point. It is also useful in solving problems involving optimization and related rates.

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