Exploring Indeterminate Limits of Rational Functions at Infinity

[LemuelUhuru]

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

The Attempt at a Solution

Here is my attempt

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

 

[vela]

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

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 $x \to \infty$.

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

 

[LemuelUhuru]

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]

Yup, that's the idea.

 

[LemuelUhuru]

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.

 

[Office_Shredder]

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.