# Tagged Questions

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### Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
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### How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
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### Calculus / exponential

Find values of a and b so that $y = a·b^x$ and the line $y = x + 2$ are tangent at $x = 0$. I tried to substitute with the zero and it seem that the $B=1$ at all time but what about the $a$?
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### Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$\int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx$$ Even though there are a lot of integrals involving the ...
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### Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $\sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!}$ Note that the inside sum gives: ...
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### Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
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### What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
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### Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
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### Approximation with 1-exponential

How come that $$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$ Is there a proof or something to understand this?
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### Comparing Powers with Different Bases Using Logarithms?

I looked all over to see if a question like this had already been answered, but I couldn't find it. So here goes: I need a general formula for comparing two (insanely huge) powers. I'm pretty sure ...
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### Choices for Integrationg by Parts

I am working on an integration by parts problem, and, it looks like I got something incorrect somewhere. I have been told to follow LIATE (L ogarithmic, I nverse trigonometric, A lgebraic, T ...
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### How many positive roots does the equation $a^x=x^a$ have?

Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've ...
### Proving the inequality $e^{-2x}\leq 1-x$
How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
### Derivative of exponential function $\frac{d}{dx}a^x$
I am trying to compute simple derivatives of simple functions, but I got stuck on $\frac{d}{dx}a^x=(\ln{a})a^x$. I suppose the proof is a simple corollary of $\frac{d}{dx}e^x=e^x$, but I am unable to ...