Can you find the limit of a complex function containing integrals and exponents?

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In summary, the limit of a complex function with integrals and exponents can be found analytically using techniques such as L'Hopital's rule, Taylor series, and integration by parts. There is no one specific method for finding the limit, and it often requires a combination of different techniques. Numerical methods such as Euler's method and Runge-Kutta methods can also be used, but may not always provide an exact solution. There are cases where the limit cannot be found analytically or numerically, such as when the function is undefined or approaches infinity. However, computer software such as Mathematica, MATLAB, and Maple can be used to find the limit accurately.
  • #1
Ackbach
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Here is this week's POTW:

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Find a real number $c$ and a positive number $L$ for which
$$\lim_{r\to\infty} \frac{\displaystyle r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{\displaystyle \int_0^{\pi/2} x^r \cos(x) \,dx} = L.$$

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  • #2
No one answered this week's POTW, which was Problem A-3 in the 2011 Putnam archive. The solution, attributed to Kiran Kedlaya and associates, follows:

[sp]
We claim that $(c,L) = (-1,2/\pi)$ works.
Write $\displaystyle f(r) = \int_0^{\pi/2} x^r\sin(x)\,dx$. Then
\[
f(r) < \int_0^{\pi/2} x^r\,dx = \frac{(\pi/2)^{r+1}}{r+1}
\]
while since $\sin(x) \geq 2x/\pi$ for $x \leq \pi/2$,
\[
f(r) > \int_0^{\pi/2} \frac{2x^{r+1}}{\pi} \,dx = \frac{(\pi/2)^{r+1}}{r+2}.
\]
It follows that
\[
\lim_{r\to\infty} r \left(\frac{2}{\pi}\right)^{r+1} f(r) = 1,
\]
whence
\[
\lim_{r\to\infty} \frac{f(r)}{f(r+1)} = \lim_{r\to\infty}
\frac{r(2/\pi)^{r+1}f(r)}{(r+1)(2/\pi)^{r+2}f(r+1)} \cdot
\frac{2(r+1)}{\pi r} = \frac{2}{\pi}.
\]

Now by integration by parts, we have
\[
\int_0^{\pi/2} x^r\cos(x)\,dx = \frac{1}{r+1} \int_0^{\pi/2} x^{r+1} \sin(x)\,dx
= \frac{f(r+1)}{r+1}.
\]
Thus setting $c = -1$ in the given limit yields
\[
\lim_{r\to\infty} \frac{(r+1)f(r)}{r f(r+1)} =
\frac{2}{\pi},
\]
as desired.
[/sp]
 

FAQ: Can you find the limit of a complex function containing integrals and exponents?

1. Can the limit of a complex function with integrals and exponents be found analytically?

Yes, it is possible to find the limit of a complex function analytically by using techniques such as L'Hopital's rule, Taylor series, and integration by parts.

2. Is there a specific method for finding the limit of a complex function with integrals and exponents?

There is no one specific method for finding the limit of a complex function with integrals and exponents. It often requires a combination of different techniques and approaches depending on the complexity of the function.

3. Can numerical methods be used to find the limit of a complex function with integrals and exponents?

Yes, numerical methods such as Euler's method and Runge-Kutta methods can be used to approximate the limit of a complex function with integrals and exponents. However, these methods may not always provide an exact solution.

4. Are there any special cases where the limit of a complex function with integrals and exponents cannot be found?

Yes, there are cases where the limit of a complex function with integrals and exponents cannot be found analytically or numerically. This can happen when the function is undefined or when the limit approaches infinity or oscillates indefinitely.

5. Can computer software be used to find the limit of a complex function with integrals and exponents?

Yes, computer software such as Mathematica, MATLAB, and Maple can be used to find the limit of a complex function with integrals and exponents. These programs have built-in functions and algorithms for handling complex mathematical expressions and can provide accurate results.

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