# Limits of Rational Functions

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→+∞$\left[\frac{f(x)}{g(x)}\right]$ = + ∞

## The Attempt at a Solution

Here is my attempt

Let f(x) = 1/x and g(x) = (1/x)+1
limx→+∞ $\left[\frac{\frac{1}{x}}{\frac{1}{x}+1}\right]$= $\frac{1}{x}$ * $\frac{x}{1}+1$ = ∞ + 1 = + ∞

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vela
Staff Emeritus
Homework Helper
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→+∞$\left[\frac{f(x)}{g(x)}\right]$ = + ∞

## 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→+∞$\left[\frac{\frac{1}{x}}{\frac{1}{\frac{1}{x}+1}}\right]$ = $\frac{1}{x}$ * $\frac{x}{1}+1$ = ∞ + 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."

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

vela
Staff Emeritus
Homework Helper
Yup, that's the idea.

• 1 person
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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Office_Shredder
Staff Emeritus