Limits As X Approaches Infinity and Negative Infinity

In summary, the conversation discusses finding the limit of three different functions as x approaches infinity and negative infinity. The first two functions, g(x) and f(x), have limits of 1/2 and 2/5 respectively, while the third function, h(x), has a limit of 9/2 as x approaches infinity and negative infinity. The conversation also briefly discusses the concept of limits and how to approach solving for them.
  • #1
dylmans
13
0

Homework Statement


Find the limit of each function
(a) as x approaches infinity and
(b) as x approaches negative infinity


Homework Equations


1. g(x)=1/(2+(1/x))

2. f(x)=(2x+3)/(5x+7)

3. h(x)=(9x^4+x)/(2x^4+5x^2-x+6)


The Attempt at a Solution


I don't know where to start.
 
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  • #2
Put a large positive and a large negative number into each. Does that suggest what the answer might be? Now can you figure out how to prove it? This typically means dividing the numerator and denominator by the dominant power.
 
  • #3
i don't exactly get what you mean. what number would i need to put in? answers for number 1 are both 1/2 and the answers for number 2 are both 2/5. the answer for 3 isn't in the book
 
  • #4
I was just suggesting you experiment numerically to get a feel for what a limit means. Try x=-100000 and x=100000. Are the answers close to what the book said? Now just start with the first one. What the limit of 1/x as x goes to either infinity?
 
  • #5
1 over infinity?
 
  • #6
dylmans said:
1 over infinity?

No, no. LIMIT 1/x as x goes to infinity. It APPROACHES an honest to God REAL number. What is that actual number? Infinity is not a number. 1/10, 1/100, 1/1000, 1/10000. What are they getting closer and closer to?
 
  • #7
they're getting closer to infinity...
 
  • #8
No, son, they are getting closer to zero. Let's rewrite them in decimal 0.1, 0.01, 0.001, 0.0001, etc.
 
Last edited:
  • #9
oh duh, ok, so what do i need to do to figure out the answer, plug in zero?
 
  • #10
Well, what's 1/(2+almost zero)? For the second one divide numerator and denominator by x. Now you have (2+almost zero)/(5+almost zero).
 
  • #11
ok, i think i get those two better. so for 3, id start by trying to factor, which i don't see anything that factors off the top of my head...
 
  • #12
Divide numerator and denominator by x^4. As x->infinity, x^4 is the large term. The rest go to zero for the same reason 1/x did.
 
  • #13
ok, i think i get it now. so the answer for number 3 would be 9/2 as it approaches infinity and the same as it approaches negative infinity?
 
  • #14
ok so the next problem is the limit as x approaches infinity for (2+x^1/2)/(2-x^1/2). the answer i got was -1.
 
  • #15
I believe that.
 
  • #16
ok, that works, thanks for the help. i'll post if i run into anymore problems
 

1. What does it mean when a limit approaches infinity?

When a limit approaches infinity, it means that the function's output is becoming larger and larger without bound as the input value (x) gets closer and closer to a certain value (infinity).

2. How do I determine the limit as x approaches infinity?

To determine the limit as x approaches infinity, you can use the following steps:

  1. Substitute infinity for x in the function.
  2. Simplify the resulting expression, if possible.
  3. If the simplified expression has a finite value, then that is the limit. If the simplified expression has no finite value (e.g. it becomes undefined or approaches infinity), then the limit does not exist.

3. How do I determine the limit as x approaches negative infinity?

The process for determining the limit as x approaches negative infinity is similar to the process for determining the limit as x approaches infinity. The only difference is that you substitute negative infinity for x instead of positive infinity.

4. Can the limit as x approaches infinity be a negative or imaginary number?

Yes, the limit as x approaches infinity can be a negative or imaginary number. This depends on the behavior of the function as x gets closer to infinity. If the function's output approaches a negative or imaginary number as x approaches infinity, then that is the limit.

5. How does the concept of "limits as x approaches infinity" relate to real world applications?

The concept of limits as x approaches infinity is often used in calculus and other mathematical fields to understand the behaviors of functions and their graphs. In real world applications, this concept can be used to model and predict the behavior of systems that involve continuously changing quantities, such as population growth, radioactive decay, or economic trends. By understanding the limits of these functions as x approaches infinity, we can gain insights into how these systems will behave in the long term.

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