Limits of Functions and Asymptotes

In summary, the conversation discusses finding discontinuities and classifying them into vertical, horizontal, and other types. It is mentioned that vertical asymptotes can be found by factoring the numerator and denominator, while horizontal asymptotes can be found by taking the limit as x approaches infinity. It is also clarified that removable discontinuities occur when the limit exists but the function is not defined at that point.
  • #1
Econometricia
33
0
1. I am concerned with finding the discontinuities of a functin say x3+3x2+2x / (x-x3)
2. I am having issues with classifying for the type of discontinuities. Finding them is not an issue.
3. Also when finding horizontal asymptotes F(x) = 4 / (2e-x +1) I understnad that the HA are found by taking the limit of the function as X--->-INF/INF , but why is one of the HA 4?

Thank You =).

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  • #2
Econometricia said:
1. I am concerned with finding the discontinuities of a functin say x3+3x2+2x / (x-x3)



2. I am having issues with classifying for the type of discontinuities. Finding them is not an issue.
Do you mean classifying them into vertical or horizontal asymptotes? The vertical asymptotes are generally the numbers that make the denominator zero, that don't also make the numerator zero. To find them, factor both the numerator and denominator. The numbers that make the denominator zero are x = -1, x = 0, and x = 1. x = 0 is NOT a vertical asymptote, because the numerator is also zero when x = 0.
Econometricia said:
3. Also when finding horizontal asymptotes F(x) = 4 / (2e-x +1) I understnad that the HA are found by taking the limit of the function as X--->-INF/INF , but why is one of the HA 4?
As x --> infinity, e-x --> 0, so the denominator --> 1, and the overall fraction --> 4.
Econometricia said:
Thank You =).
 
  • #3
Thank You for your help. I was actually meaning the discontinuities as Jump,Removable, Infinite. So far I have understood that only piece wise functions can have a Jump. Infinite limits are when the lim of F(x) as X-->A = INF/-INF. And removable is when the limit exists , but is not defined. Am I correct?
 
  • #4
Econometricia said:
Thank You for your help. I was actually meaning the discontinuities as Jump,Removable, Infinite. So far I have understood that only piece wise functions can have a Jump. Infinite limits are when the lim of F(x) as X-->A = INF/-INF. And removable is when the limit exists , but is not defined. Am I correct?
Close. A removable discontinuity occurs when [tex]lim_{x \to a} f(x)[/tex] exists (that means both one-sided limits exist), but f is not defined at a.

In your first example, I believe that there is a removable discontinuity at x = 0.
 
  • #5
Yea, that is correct. =) Thank you sir.
 

FAQ: Limits of Functions and Asymptotes

What are the different types of limits?

The three types of limits are one-sided limits, two-sided limits, and infinite limits. One-sided limits are used to determine what value a function approaches from either the left or right side. Two-sided limits are used to determine the overall behavior of a function at a specific point. Infinite limits occur when a function approaches either positive or negative infinity.

What is the definition of an asymptote?

An asymptote is a line that a graph approaches but never touches. It can be either horizontal, vertical, or oblique. Horizontal asymptotes occur when a function approaches a constant value as x approaches infinity. Vertical asymptotes occur when a function approaches infinity as x approaches a particular value. Oblique asymptotes occur when a function approaches a linear function as x approaches infinity.

How do you find the horizontal asymptote of a function?

To find the horizontal asymptote of a function, you can use the following steps:

  • Find the degree of the highest power in the numerator and denominator of the function.
  • If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0.
  • If the degree of the numerator is equal to the denominator, the horizontal asymptote is the ratio of the coefficients of the highest power terms.
  • If the degree of the numerator is greater than the denominator, there is no horizontal asymptote.

What is the difference between a removable and non-removable discontinuity?

A removable discontinuity, also known as a hole, occurs when a function has a point that is undefined, but the function can be redefined to make it continuous at that point. A non-removable discontinuity, also known as a jump, occurs when a function has a point that is undefined and cannot be redefined to make it continuous at that point. This results in a gap or jump in the graph of the function.

How do you determine the behavior of a function near an asymptote?

The behavior of a function near an asymptote can be determined by evaluating the limit of the function as x approaches the asymptote from both sides. If the limit from the left and right sides are equal, the asymptote is a horizontal asymptote. If the limit from the left side is negative infinity and the limit from the right side is positive infinity, the asymptote is a vertical asymptote. If the limit from both sides is infinity, the asymptote is an oblique asymptote.

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