# Tagged Questions

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### Why is true? $\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}$

$$\begin{array}{l}a,b > 0\\\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}\\\end{array}$$ I asked already a similar question, but I'm still not sure what ...
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### Limit involving $\frac{0}{0}$ and $\sin 2x$

I'm currently working with the limits: $$\frac{\sin(2x)}{e^x-1}$$ for $x\to 0$ and $x\to \infty$ how can this be solved? I tried to reformulate $\sin(2x)$ to $2\sin x\cos x$ Using L'Hõpital's ...
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### find the limit of $\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$

$$a,b \gt 0$$ $$\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$$ So, I know that if x is $x \in \mathbb{Z}$ then the limit is $a\over [b]$ I couldn't figure out the ...
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### Limit problem function

Prove that the statement is wrong. If $\lim f(x)$ exists but $\lim g(x)$ does not exist, then $\lim f(x)g(x)$ does not exist either.
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### Trying to prove a limit of a simple multi case function

Full disclosure: This is my attempt at solving a homework assignment Another full disclosure: This is my first time using LaTeX, so pardon any errors We were asked to prove / disprove the existance ...
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### Finding the limit of $\mathop {\lim }\limits_{x \to 0} \frac{x}{{\ln (5x + 1)}}$

$$\lim_{x \to 0^+} \frac{x}{\ln (5x + 1)} = {1 \over 5}$$ First, What I tried to do is dividing by $x$, but it didn't work out. By the way, It's a common approach to find a limit. Why is it failing ...
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### Prove the limit $\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$

Prove the following limit: $$\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$$ I can use limits arithmatic, squeezing principle, "well-known" limits etc.. We didn't learn Lopital law so I ...
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### prove the following function limit statement

Let $f$ be monotone ascending (for every $x\le y, ( f(x) \le f(y))$ in the segment $(a,b)$ prove that: $\lim_{x\to x_0^+} f(x) = \inf_{x>x_0} f(x)$ thats the question. a hint i can use ...
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### A simple question on limits

Is it true that $$\lim_{x\to+\infty} \mathbb{I}_{S=\{z\mid e^{-z}>0, z\in\mathbb{R}\}}(x) = 1,$$ where $\mathbb{I}_{S}(x)$ is an indicator function for $x\in S$?
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### Proving a limit by Cauchy definition

for $a>1$: $$\mathop {\lim }\limits_{x \to \infty } \frac{{{a^x}}}{x} = \infty$$ So, by the definition of Cauchy for limits, for any $M>0$ I need to find a $D>0$ such that: $x>D$ ...
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### what is the limit of $\displaystyle \lim_{x\to 0} \frac{x}{a}\cdot \lfloor{\frac{b}{x}\rfloor}$

find the limit of: $\displaystyle\lim_{x\to 0} \frac{x}{a}\cdot \lfloor{\frac{b}{x}\rfloor}$ $a,b>0$ i know that: $\lfloor\frac{b}{x}\rfloor\le \frac{b}{x} + 1$; and ...
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### prooving the limit of a function in epsilon,delta, and sequences

Hi I need to prove the following limit of a function: I need to show the proof in two ways, one in epsilon and delta and the other with the sequence the diverge to infinity. I'm having difficulties ...
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### How prove this $\displaystyle\lim_{\tau\to t}f(t,\tau)=\frac{1}{2\pi}\frac{x'_{1}(t)x''_{2}(t)-x'_{2}(t)x''_{1}(t)}{[x'_{1}(t)]^2+[x'_{2}(t)]^2}$

Question: let $$j_{1}(t)=\sum_{p=0}^{\infty}\dfrac{(-1)^p}{p!(1+p)!}\left(\dfrac{t}{2}\right)^{1+2p}$$ ...
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### The slope of the tangent line

I don't know how to solve these three, especially the first and the second one: $1.$ $x=0$ means upper limit also equal to $0$? what should I do to deal with the $(t^2+\pi^2)^{1/2}$? $2.$ Find the ...
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### Can someone help with this limit? How?

I know what is the solution, but I don't know how to calculate it without l'Hôpital's rule: $$\lim \limits_{x \to 1 }\frac{\sqrt[3]{7+x^3}-\sqrt{3+x^2}}{x-1}$$
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### What is a coordinate shifting?

I need to find the limit: $$\lim_{(x,y,z)\to(1,3-1)}\frac{(x-1)(y-3)+(z+1)^2}{(x-1)^2+2(y-3)^2+3(z+1)^2}.$$ A hint written below says: ...
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### Limit on both inside and the range of an Integral

$$\lim_{n\rightarrow \infty }\int_{0}^{kn}e^{tx}\frac{1}{k}\left ( 1- \frac{x}{kn}\right )^{n-1} \, \mathrm{d}x$$ Can I first show $$\frac{1}{k}\left ( 1- \frac{x}{kn}\right )^{n-1}$$ converges on ...
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### Take limit of an Integral (both the limit and the function inside)

$$\mathop {\lim }\limits_{n \to \infty } \int\limits_a^{bn} {{f_n}(x)g(x)dx}$$ I am stuck now, but I can show f_n is bounded in the region (a,bn) I tried to to this but I have no idea if this is ...
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### Limit of $n/\ln(n)$ without L'Hôpital's rule

I am trying to calculate the following limit without L'Hôpital's rule: $$\lim_{n \to \infty} \dfrac{n}{\ln(n)}$$ I tried every trick I know but nothing works. You don't have to prove it by ...
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### Evaluate limit of integral of sequence of function

Evaluate $$\lim\limits_{n\to \infty}\int_{0}^{1}\frac{\sqrt{n}(e^{-x/n}-1)}{x}dx.$$
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### Prove limit's value.

Using the definitions of limits and derivatives, for a function $f$ with $\lim_{x\to \infty} f'(x)=0$ prove that: $$\lim_{x \to \infty}( f(x+1)-f(x) )=0\text{.}$$
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### Prove that f*g is differentiable at x0.

Let f,g : R-->R. Let f(x0) = 0, f(x) differentiable at x0 and g(x) continuous at x0. I need to prove that f*g is differentiable at x0. Any ideas of hints about how to begin? *continuousity doesnt ...
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### Finding points on graph with tangent lines perpendicular to a line

Find all points $(x,y)$ on the graph of $y=\frac{x}{x-3}$ with tangent lines perpendicular to the line $y=3x-1.$ My thoughts on this problem: First I should find the slope of the given line ...
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### prove that: $\lim_{x \to \infty} [f(x+1)-f(x)] = 0$ just by using definitions of limit and definition of derivative.

Let f and it is given that $\lim_{x\to \infty} f '(x) = 0$. I have to prove that: $\lim_{x\to \infty} [f(x+1)-f(x)] = 0$ just by using definitions of limit and definition of derivative. I have no ...
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### Confusion with this definition of the derivative

This function is from my text: $$p(\theta) = \sqrt{13\theta}$$ It states that the derivative of the function $p(\theta)$ with respect to the variable $\theta$ is the function $p'$ whose value at ...
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### Finding the limit of a negative infinity function.

Studying for a midterm: Let $f(x)=\frac{2x}{(2x-1)^2}$ Then $\lim_{x->-\infty} f(x)$ is: Now keep in mind I'm shaky on how to do infinity limits. I have $f(x)=\frac{2x}{(2x-1)^2}$ Remove x by ...
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### Functions and derivatives using limits

For a calculus project, I need to create a function and find its derivative using the limit definition of derivatives. I could use a function from this list, but I want to know how I could come up ...
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### Why does this limit work?

Let $h(x)= (1+1/x)^x$ and $g(x)$ be another function. Now suppose $\lim\limits_{x \to \infty} g(x)= \infty$. Then $\lim\limits_{x \to \infty} h(g(x))$ =$\lim\limits_{x \to \infty} h(x)=e$. I would ...
Is there any representation of the exponentil function as infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. \mathrm ...
### Show $g(x)=\sqrt{x}$ is continuous at x=4
I just really need to make sure that I am understanding the process for doing these. Scratch work: We have $|\sqrt{x}-\sqrt{4}| = |\sqrt{x}-2|= |\frac{x-4}{\sqrt{x+2}}|= \frac{|x-4|}{|\sqrt{x}+2|}$. ...