Limit problem involving x to the power of a function

In summary, the problem is to calculate the limit of x^(1/x^2) as x approaches infinity. The first attempt was to rewrite the function as a composition, but this did not work. The next attempt was to manipulate the function algebraically, but it only resulted in going in circles. Another method was suggested, which involves rewriting the function as x^(lim(x->infinity) 1/x^2). This seems plausible, but it is not clear if it is a valid operation. Another suggestion was to take the limit of ln(y), where y = x^(1/x^2).
  • #1
Wormaldson
21
0

Homework Statement



Calculate the limit [tex]\lim_{x\rightarrow +\infty} x^{\frac{1}{x^2}}[/tex]

2. The attempt at a solution

At first I was tempted to rewrite the function as a composition and go from there. I soon realized that this wouldn't work as the term being raised to the power of a function of x was a different function of x; to my knowledge, decomposing functions in this way will only work for functions like [tex]a^{f(x)}[/tex]So no luck there.

The next thing I tried was manipulating the function algebraically; unfortunately I just ended up going in circles and wasn't able to get the function into a more useful form.

I came up with another method that seems superficially plausible, but I'm not sure if it's actually valid: that is, rewriting the problem as [tex]x^{\lim_{x\rightarrow +\infty} {\frac{1}{x^2}}}[/tex]Intuitively, this seems sensible; the limit at infinity of the exponent is 0 and any x > 0 raised to the power of 0 is 1. The course book mentions that you can take a limit "inside" a continuous function (the exact wording). Obviously the function in question isn't continuous on it's entire domain, but it is continuous for all x > 0. Unfortunately, however, the course book doesn't go into any greater detail, and the textbook doesn't address limits of this type at all. If taking the limit "inside" the function in such a manner is a valid operation, then I guess my problem is solved. If not, then I could use a suggestion as to what to try next.

As always, any help is much appreciated.
 
Physics news on Phys.org
  • #2
If ##y = x^\frac 1 {x^2}##, try taking the limit of ##\ln y##.
 
  • #3
Using that x=eln(x), the function can be written in the form [tex](e^{\ln(x)}) ^{\frac{1}{x^2}}=e^{\ln(x)\frac{1}{x^2}}[/tex]

ehild
 

FAQ: Limit problem involving x to the power of a function

1. What is a limit problem involving x to the power of a function?

A limit problem involving x to the power of a function is a mathematical concept where the value of a function at a certain point is determined as the input value, or x, approaches a given value, typically represented as a. This is denoted by the notation lim x→a f(x) and is used to analyze the behavior of a function near a specific point.

2. How do I solve a limit problem involving x to the power of a function?

To solve a limit problem involving x to the power of a function, you first need to substitute the value of x into the function. Then, simplify the expression as much as possible by using algebraic manipulations. If the resulting expression is still indeterminate, you may need to use techniques such as L'Hopital's rule or factoring to further simplify the expression. Finally, plug in the value of a to determine the limit value.

3. What are the common types of limit problems involving x to the power of a function?

The common types of limit problems involving x to the power of a function include polynomial functions, rational functions, exponential functions, and logarithmic functions. Each type may require different techniques to solve, so it is important to understand the properties and behaviors of each type.

4. Why are limit problems involving x to the power of a function important?

Limit problems involving x to the power of a function are important because they allow us to understand the behavior of a function near a specific point. This can help in determining the continuity, differentiability, and other properties of the function. It is also a fundamental concept in calculus and is used in many real-life applications, such as in physics, engineering, and economics.

5. Are there any common mistakes to avoid when solving limit problems involving x to the power of a function?

Yes, there are some common mistakes to avoid when solving limit problems involving x to the power of a function. These include forgetting to simplify the expression before plugging in the value of a, using incorrect techniques for different types of functions, and not understanding the properties of the function. It is important to carefully follow the steps and use the appropriate techniques to avoid making mistakes in solving these types of problems.

Similar threads

Find limit of multi variable function
Replies
10
Views
1K
Calculate the limit cos(x)/sin(x) when x approaches 0
Replies
12
Views
2K
Can this function still be a constant function?
Replies
11
Views
539
Proving piecewise function is k-differentiable
Replies
5
Views
883
Limit and Integration of ##f_n (x)##
Replies
8
Views
1K
Non-Differentiable Function proof
Replies
8
Views
679
Determine the limit of 2^x/x^2 as x approaches infinity....
Replies
26
Views
2K
Why is this valid? Continuity of Functions Problem
Replies
3
Views
497
Proof by induction for rational function
Replies
7
Views
825
On the ratio test for power series
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top