Finding Limit x -> Infinity of y = 2x^2/(a+2x)

Thread starter forty
Start date Mar 9, 2009
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Infinity Limit

In summary, the purpose of finding the limit of a function as x approaches infinity is to understand its behavior at extremely large values of x and make predictions about its long-term behavior. To find the limit, we substitute infinity for x and simplify the resulting expression. The limit can be either a finite number or infinity/negative infinity. For the function y = 2x^2/(a+2x), the limit as x approaches infinity is infinity. It can also be negative, depending on the value of a. The value of a affects the function's behavior and can result in a horizontal or slant asymptote, or a limit of 0.

Mar 9, 2009

forty

I need to find the limit x -> infinity of the following:

y = x ( (2x/a) / (1 + (2x/a)) )

Simplifying..

y = x ( (2x/a) / ((a + 2x)/a) )

y = x ( 2x / (a + 2x) )

y = 2x^2 / (a + 2x)

Is this even right in the first place? because I have no idea how to evaluate the lim x -> infinity.

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Mar 9, 2009

HallsofIvy

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If you were to divide the numerator by the denominator what would you get?

1. What is the purpose of finding the limit of a function as x approaches infinity?

The limit of a function as x approaches infinity helps us understand the behavior of the function at extremely large values of x. It gives us a better understanding of the function's overall behavior and can help us make predictions about its long-term behavior.

2. How do you find the limit of a function as x approaches infinity?

To find the limit, we substitute infinity for x in the function and simplify the resulting expression. If the resulting expression approaches a finite number, then that is the limit. If the resulting expression approaches infinity or negative infinity, then the limit does not exist.

3. What is the limit of y = 2x^2/(a+2x) as x approaches infinity?

The limit of y = 2x^2/(a+2x) as x approaches infinity is infinity. When we substitute infinity for x, the resulting expression becomes 2x^2/infinity, which simplifies to infinity. This means that as x gets larger and larger, the value of the function also gets larger and larger.

4. Can the limit of a function as x approaches infinity be negative?

Yes, the limit of a function as x approaches infinity can be negative. It is possible for the resulting expression to approach negative infinity when we substitute infinity for x. This means that as x gets larger and larger, the value of the function becomes more and more negative.

5. How does the value of a in the function y = 2x^2/(a+2x) affect the limit as x approaches infinity?

The value of a in the function y = 2x^2/(a+2x) affects the limit as x approaches infinity by changing the behavior of the function. If a is a positive number, the function will have a horizontal asymptote at y = 2. If a is a negative number, the function will have a slant asymptote at y = 2x. If a is equal to 0, the limit as x approaches infinity will be 0.