Solve a limit with the form (x^n)(e^ax) when x approaches to infinity?

In summary, to solve a limit with the form (x^n)(e^ax) when x approaches to infinity, you do not need to use L´Hopital rule. It is enough to show that both (x^n)(e^ax) and (e^ax) have the same limit. Additionally, it is known that n and a are both greater than 0.
  • #1
Wilfired
2
0
solve a limit with the form (x^n)(e^ax) when x approaches to infinity?

Well, my question is how to solve a limit with the form (x^n)(e^ax) when x approaches to infinity using L´Hopital rule??
I made a try, transforming the limit to (x^n)/(e^-ax), and using L´Hopital repeatedly, gives me something like this:
(nx^n-1/(-ae^ax), (n(n-1)x^n-2/(a^2e^-ax)...
So, the question, if this is correct (although i don't think so), is how i can simplify this??
If i´m wrong (which is more probably), please don't be so hard with me, hehe :)
Thanks for the answers!
 
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  • #2


You do not need L´Hopital rule. Show (x^n)(e^ax) and (e^ax) have the same limit.
 
  • #3


Is it known that n and a are > 0?
 
  • #4


haruspex said:
Is it known that n and a are > 0?

Yes!
 

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a particular value.

2. What is the form (x^n)(e^ax) when x approaches to infinity?

The form (x^n)(e^ax) when x approaches to infinity is an indeterminate form, meaning that the value of the expression cannot be determined just by looking at its components. It is often encountered when taking limits of functions involving exponential and polynomial terms.

3. How do you solve a limit with the form (x^n)(e^ax) when x approaches to infinity?

To solve a limit with this form, we use the L'Hopital's rule, which states that the limit of a quotient of two functions is equal to the limit of their derivatives, provided that the limit of the denominator is not equal to zero. In this case, we take the natural logarithm of the expression and then apply the rule, simplifying the expression until we are left with a determinate form.

4. What is the significance of taking the natural logarithm when solving a limit with the form (x^n)(e^ax) when x approaches to infinity?

Taking the natural logarithm of the expression allows us to simplify it into a determinate form. This is because the natural logarithm of an exponential function is equal to its exponent. By using this property, we can simplify the indeterminate form and evaluate the limit using L'Hopital's rule.

5. Are there any other methods for solving a limit with the form (x^n)(e^ax) when x approaches to infinity?

Yes, there are other methods such as using substitution or algebraic manipulation. However, these methods may not always work for every expression. L'Hopital's rule is the most efficient and reliable method for solving limits with this form.

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