Find the following limit:$$\lim\limits_{x\to \:4}\frac{\sqrt{x+5}-3}{x-4}$$ I tried to multiply by the conjugate and it did not work.

I’m asked to prove the question above. I need to show that if $(a_n)$ is an increasing sequence of integers then: $\lim_{n\to\infty}\left(1+\frac{1}{a_n}\right)^{a_n}=e$ I was thinking of showing that $\lim_{n\to\infty}(a_n)=\infty$ and then, by definition, I can say that there exists a natural number $N$, such that for every $n>N, a_n>0$ and so $(a_n)^\infty_{n=N}$ is a sub-sequence […]

$$\lim_{x \to 1} \frac {2x+5}{x^{2}-4x+3}$$ We have $0$ in denominator i don’t know how to use l’Hoptial rule and i think we can’t use the rule of polynomial functions

I am wondering about a multivariable limit, and in particular, is it ever valid to use L’hospital rule. For example, I am working on $$ \lim_{(x,y) \to (1,1)} \frac{x^3-y}{x-y}$$ This is what I have done, let $$f(x,y)=\frac{x^3-y}{x-y}$$ $f(x,0) \rightarrow 1$ as $(x,y) \rightarrow (1,1)$ and similiary $f(0,y) \rightarrow 1$ as $(x,y) \rightarrow (1,1)$ Okay now […]

Problem: $f'(x)$ is a continuous function for $[0, 1]$. Show that $$\lim_{n \to \infty}n \left( \frac{1}{n}\sum_{i=1}^n {f\left(\frac{i}{n}\right)-\int_0^1f(x)dx} \right)=\frac{f(1)-f(0)}{2}$$ I tried to use the definition of the definite integral to change it to a limit but it doesn’t seem to work. And I also wonder why a continuous condition for $f'(x)$ was given. I thought it […]

This question already has an answer here: Prove that $( 1 + n^{-2}) ^n \to 1$. 8 answers

$$\large f(x)= \lim_{n\rightarrow \infty}\left( \dfrac{n^n(x+n)\left( x+\dfrac{n}{2}\right)\left( x+\dfrac{n}{3}\right)… \left( x+\dfrac{n}{n}\right)}{n!(x^2+n^2)\left( x^2+\dfrac{n^2}{4}\right)\left( x^2+\dfrac{n^2}{9}\right)…\left( x^2+\dfrac{n^2}{n^2}\right)}\right)$$ $x\in R^+$ Find the coordinates of the maxima of $f(x)$. My Work: Is the method correct? Is there an easier way?

Is it possible to use $\delta=\min(1,\frac\varepsilon c)$ in the following exercise? Thanks in advance!! $$\lim_{x\to 1}(x^2+4x)=5.$$ To make my question clear, do we have a right to choose the limit constant a as $\delta$?

The Harmonic number $H_n$ is defined as $H_n=\sum_{t=1}^n\frac{1}{t}$. I wish to compute $\lim_{n\to\infty}\frac{k}{n}(H_n-H_k)$ where $k$ is a function of $n$ (which can be a constant function, e.g. $k=3$). Moreover, I wish the computation to be as simple as possible and self-sufficient (which might already be too much to ask). If a general solution is not […]

I was inspired by this <link> on page 10-11, where it was shown that $$\lim_{x\to 1}\left(4+x-3x^{3}\right)=2.$$ I wonder if it’s possible to show this way without choosing a good value of $\delta>0$ (for example $\delta\leq 1$ in the link). Since $$\left | (4 + x-3x^{3})-2 \right |=\left | x-1 \right |\left | -3x^{2}-3x-2 \right |,$$ […]

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