The limit as x approaches infinity

Then the limit becomes (3- 2/x)/ (9+ 7/x). As x goes to infinity, 2/x goes to 0 so the limit is just 3/9= 1/3.
  • #1
Goldenwind
146
0

Homework Statement


(3x-2) / (9x+7)

As x approaches infinity.

The Attempt at a Solution


I know the procedure, but am then stuck:
- Rearrange
- Plug in infinity for x
- Evaluate

Tried breaking it into 3x / (9x+7) - 2 / (9x+7)
The second thing would go to zero.
The first, no clue.

Maybe multiplying the original problem by something? Unsure what.

(Note: This is to help a friend. I myself am going to bed now, but they will be checking for replies. They're not allowed to use L'Hopital's rule, as it wasn't taught)
 
Last edited:
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  • #2
factor out x from both num and den!
and plug in the infinity value without thinking
 
  • #3
Goldenwind said:

Homework Statement


(3x-2) / (9x+7)

As x approaches infinity.

The Attempt at a Solution


I know the procedure, but am then stuck:
- Rearrange
- Plug in infinity for x
- Evaluate
No, you don't "know the procedure". Only in the most trivial limits (limits of continuous functions where, by definition, the limit is the value of the function, can)you "plug" the target point into the formula. And certainly you can't "plug infinity" into this formula because it is only defined for real numbers and "infinity" is not a real number.

Tried breaking it into 3x / (9x+7) - 2 / (9x+7)
The second thing would go to zero.
The first, no clue.

Maybe multiplying the original problem by something? Unsure what.

(Note: This is to help a friend. I myself am going to bed now, but they will be checking for replies. They're not allowed to use L'Hopital's rule, as it wasn't taught)
It is true, however, that limit, as "x goes to infinity", of 1/x is 0 and 0 is easy to use. In order to change from "x" to "1/x" divide both numerator and denominator by x (that is the same as rootX's "factor x from numerator and denominator").
 
Last edited by a moderator:

What is the concept of the limit as x approaches infinity?

The limit as x approaches infinity is a mathematical concept used to describe the behavior of a function as the input value (x) approaches an infinitely large value. It is a way of understanding what happens to the output of a function as the input becomes larger and larger.

How is the limit as x approaches infinity calculated?

To calculate the limit as x approaches infinity, you can simply plug in a large value for x (such as 100 or 1000) and see what the output of the function is. If the output approaches a specific number, then that number is the limit. If the output becomes larger and larger without approaching a specific number, then the limit does not exist.

What is the difference between a one-sided and two-sided limit as x approaches infinity?

A one-sided limit as x approaches infinity only considers the behavior of the function as x becomes infinitely large in one direction (either positive or negative). A two-sided limit takes into account the behavior of the function as x approaches infinity from both positive and negative directions.

Why is the limit as x approaches infinity important in calculus?

The limit as x approaches infinity is important in calculus because it allows us to understand the behavior of a function at extreme values. It helps us to determine the continuity of a function, identify vertical and horizontal asymptotes, and evaluate the convergence or divergence of infinite series.

What are some real-world applications of the limit as x approaches infinity?

The concept of the limit as x approaches infinity has many real-world applications, such as in physics, economics, and engineering. For example, it can be used to model the growth of a population over time, the decay of radioactive materials, and the convergence of algorithms in computer science.

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