Limits of Rational Functions

In summary, the conversation was about finding examples of functions f and g with specific properties and limits, such that the limit of f(x)/g(x) as x approaches infinity is positive infinity. The attempt at a solution involved using f(x) = 1/x and g(x) = (1/x)+1, but it was pointed out that g(x) does not approach zero in the limit. The student then asked for a more systematic approach to solving this type of problem, and the expert suggested making f(x) = 1/F(x) and g(x) = 1/G(x), where F(x) and G(x) go to infinity but G(x) goes to infinity faster.
  • #1
Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my solution void.

Homework Statement



Suppose f(x) →0 and g(x) → 0 as x → +∞. Find examples of functions f and g with these properties and such that:
a. limx→+∞[itex]\left[\frac{f(x)}{g(x)}\right][/itex] = + ∞

The Attempt at a Solution



Here is my attempt

Let f(x) = 1/x and g(x) = (1/x)+1
limx→+∞ [itex]\left[\frac{\frac{1}{x}}{\frac{1}{x}+1}\right][/itex]= [itex]\frac{1}{x}[/itex] * [itex]\frac{x}{1}+1[/itex] = ∞ + 1 = + ∞
 
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  • #2
LemuelUhuru said:
Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my solution void.

Homework Statement



Suppose f(x) →0 and g(x) → 0 as x → +∞. Find examples of functions f and g with these properties and such that:
a. limx→+∞[itex]\left[\frac{f(x)}{g(x)}\right][/itex] = + ∞

The Attempt at a Solution



Here is my attempt

Let f(x) = 1/x and g(x) = (1/x)+1
First problem, g(x) doesn't go to 0 in the limit as ##x \to \infty##.

limx→+∞[itex]\left[\frac{\frac{1}{x}}{\frac{1}{\frac{1}{x}+1}}\right][/itex] = [itex]\frac{1}{x}[/itex] * [itex]\frac{x}{1}+1[/itex] = ∞ + 1 = + ∞
I'm not sure what you did here because you have f(x) in the numerator but 1/g(x) in the denominator, and then you did some funky "algebra."
 
  • #3
Hi Vela,

I realize now that g(x) doesn't go to zero, that was a huge error and I adjusted the original attempt with respect to 1/g(x), although that despite it being irrelevant now, however, I am still unsure how to approach this problem generally. I imagine that I am suppose to obtain to two functions both f(x) and g(x) who limit approaches zero and when divided result in a/0 to give me positive infinity?

Thank you,

Lemuel
 
  • #4
Yup, that's the idea.
 
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  • #5
I understand, however, right now I'm simply guessing what rational expressions would allow me to do this, is there a more systematic approach to solving this kind of problem?

Also, it seems difficult to obtain a/0 since the limit of f(x) and g(x) as x approaches infinity is zero and any two functions where a/∞ cancels out my constant when x is subtituted for infinity.
 
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  • #6
All you're looking for is for g(x) to be something that goes to zero really fast, and f(x) to be something that goes to zero slower.

If you're not sure how to do that, make f(x) = 1/F(x) and g(x) = 1/G(x) where F(x) and G(x) go to infinity, but G(x) much faster.
 

1. What are rational functions?

Rational functions are mathematical expressions that are made up of polynomials (algebraic expressions with variables and coefficients) divided by other polynomials.

2. What are the limits of rational functions?

The limits of a rational function are the values that the function approaches as the input (x) approaches a certain value. This value can be a real number or infinity.

3. How do you find the limits of rational functions?

To find the limits of a rational function, you can use algebraic methods such as factoring and simplifying the expression. You can also use the properties of limits, such as the limit laws and the Squeeze Theorem.

4. What are the types of limits for rational functions?

There are three types of limits for rational functions: horizontal asymptotes, vertical asymptotes, and holes. Horizontal asymptotes are the values that the function approaches as x goes to infinity or negative infinity. Vertical asymptotes are the values where the function is undefined. Holes are points where the function is undefined, but can be "filled in" to make the function continuous.

5. Why are limits of rational functions important?

Limits of rational functions are important because they help us understand the behavior of a function near certain input values. They also help us determine if a function is continuous and if it has any vertical or horizontal asymptotes, which are important in graphing and understanding the overall shape of a function.

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