Articles of limits

Integral $\int_1^2 \frac1x dx$ with a Riemann sum.

How do you find the $$ \int \dfrac{1}{x} dx$$ by using the idea of a limit of a Riemann sum on the interval [1,2]? I tried splitting the interval into a geometric progression and evaluating the Riemann sum, but i cant simplify the expression at this stage.

The limit $\lim_{n\to \infty}\frac{T_n(n)}{e^n}$ where $T_n(x)$ is the Taylor polynomial of $e^x$

This question already has an answer here: Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 8 answers

Need a tip/hint evaluating a limit

I have the following limit: $$\lim_{x\rightarrow\infty}\left(1+\frac{a}{x^{1/2+\epsilon}}\left(1-\exp\left(-\frac{b}{x^{1/2+\epsilon}}\right)\right)\ln\left(\frac{a}{x^{1/2+\epsilon}}\right)\right)^x$$ where $0<b<a$. I care for the case where $\epsilon>-1/2$. I suspect that for $\epsilon>0$ this limit evaluates to 1, and for $-1/2<\epsilon\leq0$ it evaluates to 0. However, I am having hard time evaluating this. I have tried taking the log of the expression (moving the $x$ in the exponent […]

Nonconvex set converging to a convex set despite holes

I’m looking at the example in Figure 4-7 of “Variational Analysis” (Rockafellar and Wets). Basically, there’s a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence eventually converges to the set $C$ (with the same shape but without holes) as long as the holes get finer and finer and thus vanish […]

If $\lim\limits_{x \to \infty} f(x)$ is a finite real number and $f''(x)$ is bounded, then $\lim\limits_{x \to \infty} f'(x) = 0$

This question already has an answer here: If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ 3 answers

Finding the limit of a quotient

I am trying to find the limit of $(x^2-6x+5)/(x-5)$ as it approaches $5$. I assume that I just plug in $5$ for $x$ and for that I get $0/0$ but my book says $4$. I try and factor and I end up with $(25-30+5)/(5-5)$ which doesnt seem quite right to me but I know that […]

How to solve $\lim_{x\to1}\left$ without using L'Hospital's rule?

$$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$$ How can I solve this without using the L’Hopital’s rule? Any tips or hints would be greatly appreciated. I tried using the substitution $x=e^y$, but to no avail.

$\lim_{\lambda \to \infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t} $

For a continuous function, $f:[0,b] \to \Bbb{R}$ show that: $$ \lim_{\lambda\to\infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t}\,dt = \frac{\pi}{2}\,f(0) $$ I know it has something to do with the Riemann-Lebesgue lemma about Fourier series, but $\frac{f(t)}{t}$ is not an integrable function. I tried to define $g(t) = \frac{2f(t)\sin(\frac{t}{2})}{t} $ which tends to $f(0)$ as $t\to0$ and $g(t)$ […]

l'Hôpital guidelines

Suppose we want to evaluate $$\lim_{x \to 0} \; x \log (x)$$ If we write this in the form $x/(1/\log(x))$, then l’Hôpital’s rule does not work, but if we write it as $\log(x)/(1/x)$, it does. Is there any sort of general guideline to choosing the numerator and denominator?

Stuck on Infinite L'hopitals

I have been trying forever to figure out this problem, but I seem to get stuck in an infinite L’hopitals loop. See the question below: Find the value of the positive constant c such that: $\lim_{x \to \infty}(\frac{x+c}{x-c})^x=10$ After rearrainging the problem a few times (mainly because of other indeterminate forms) I get stuck here: […]