Find Value of k for Limit: x^3-6 / x^k+3

Thread starter starchild75
Start date Jan 20, 2008
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Limit Value

In summary, the value of k in a limit expression helps determine the behavior of the function as x approaches infinity or negative infinity, and whether the limit exists or not. To find the value of k, the limit expression must be evaluated and compared with the given function. The value of k must be a real number that satisfies the limit expression, and if it cannot be found, it means the limit does not exist or the function does not approach a specific value. The value of k also determines the end behavior of the function, with even k resulting in the same end behavior as a polynomial function with the same degree as the exponent of x, and odd k resulting in the opposite end behavior.

Jan 20, 2008

starchild75

Homework Statement

find the value of k such that the limit exists. lim xgoes to infinity (x^3-6)/(x^k+3).

Homework Equations

The Attempt at a Solution

I multiply both sides by 1/x^3 I get 1 in the numerator and x^k/x^3 in the denominator. I don't know what to next.

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Jan 20, 2008

Dick

Science Advisor

Homework Helper

x^k/x^3 is x^(k-3). Consider the cases of k>3, k<3 and k=3.

1. What is the purpose of finding the value of k for this limit?

The value of k in this limit helps determine the behavior of the function as x approaches infinity or negative infinity. It is also used to determine if the limit exists or not.

2. How do I find the value of k for this limit?

To find the value of k, you need to set up the limit expression and evaluate it as x approaches infinity or negative infinity. Then, you can compare the resulting expression with the given function to determine the value of k.

3. Can the value of k be any real number?

No, the value of k must be a real number that satisfies the given limit expression. It cannot be any arbitrary number.

4. What happens if the value of k is not found?

If the value of k cannot be found, it means that the limit does not exist or the given function does not approach a specific value as x approaches infinity or negative infinity.

5. How is the value of k related to the end behavior of the function?

The value of k determines the end behavior of the function. If k is even, the function will have the same end behavior as a polynomial function with the same degree as the exponent of x. If k is odd, the end behavior will be the opposite of a polynomial function with the same degree as the exponent of x.