Questions on the evaluation of limits.

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0
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1answer
14 views

Help finding slope of a secant line

The point $P(5,-2)$ lies on the curve $y=\large\frac{2}{4-x}$. (a). If $Q$ is the point $(X,\large\frac{2}{4-x})$, find the slope $M_{pq}$ of the secant line $PQ$ (correct to six decimal places). ...
0
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0answers
22 views

Limit of Mathieu function near the discontinuous point

Consider the Mathieu characteristic function, which is a piecewise function. The discontinuity happens at integer number. ...
0
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1answer
46 views

Show that $f^{[n]}(0)=0$ for all $n=0,1,2…$

Let $$f(x)=\left\{ {\matrix{ {{e^{ - {1 \over {{x^2}}}}},x \ne 0} \cr {0,x = 0} \cr } } \right.$$ Show that $f^{[n]}(0)=0$ for all $n=0,1,2,\cdots$ The proof: First note that for ...
5
votes
1answer
46 views

To find a trigonometric limit without Wallis' integrals

What is the limit $$ \lambda =\lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx}$$ I would like to find it without Wallis' integral formula: I mean without evaluating the closed ...
1
vote
3answers
54 views

Limit at Infinity Problem.

$$ \lim_{x\to\infty} \frac{2x^{1/2}+3x^{1/3}+5x^{1/5}}{(3x-2)^{1/2}+(2x-3)^{1/3}} $$ I tried to divide the expresion with the highest power of $x$, but the problem is with the denominator.
1
vote
2answers
40 views

How to graph functions without a calculator?

How do you graph a function such as $$f(x)=\frac{x^2+3x+2}{x+1}$$ and find its limits $\lim_{x\to-1^-}f(x)$, $\lim_{x\to-1^+}f(x)$, $\lim_{x\to-1}f(x)$? Thank you!
1
vote
1answer
39 views

Calculating multi-variable limit.

I am struggling to find a way to approach this limit $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I would greatly appriciate if You could explain to me how to solve it or at least show ...
1
vote
1answer
22 views

Limit problem using sequential criteria for limits

$$\lim(n+n^2\log \frac{n}{n+1})= \frac12$$ How? In the text book it is simply said that this happens by Sequential criteria of limits. I don't get it.
1
vote
1answer
74 views

What is the limit regarding $a$

What is the limit of : $$ \lim_{x\to 0} \frac{\sin(ax) - \ln(1-2x)}{e^{ax}-1-2x-2x^{2}}$$ I did this with Maclaurin, because my exam is about solving these with MacLaurin. Gave $$\lim_{x\to 0} ...
2
votes
1answer
66 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
0
votes
1answer
18 views

multivariate limits proof

Could anyone help me with this proof? Given a map $f : \mathbb R^m \to\mathbb R^n,$ prove that $\lim_{x \rightarrow a} f(x) = L$ holds if and only if $$\lim_{x ...
2
votes
1answer
62 views

What is $\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$? [duplicate]

What is the result of the following limit? $$\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$$ since $$\sum_{k=0}^{2n}{2n \choose {k}}=2^{2n}=4^n$$ then $$\frac{4^n}{2n+1}\leq{2n \choose {n}}\leq 4^n$$ ...
3
votes
7answers
108 views

How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?

Show that $$\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i} =0 $$ I've proved that this sequence converges (it is bounded and decreasing). NOW, I need to find a sequence that is bigger than this one and ...
3
votes
1answer
83 views

When may we ignore the limits of integration?

When we try to evaluate an integral such as, say $$\int_a^b{f(x)dx}$$ there is often the case that we can analytically find $$\int{f(x)dx}$$ a little faster (imagine leaving away the evaluation ...
1
vote
2answers
29 views

sequences/limits

The book I use is Jon Rogawski, multivariable calculus, chapter 1, question 39: evaluate lim {n (sin 1/n)}, for n→∞. the student solution manual gives a fairly detailed explanation, it says: 1): lim ...
-3
votes
1answer
55 views

Epsilon Delta definition to prove $\lim_{x\to a} x^{1/3} = a^{1/3}$ [on hold]

prove $$\lim_{x\to a} x^{1/3} = a^{1/3}$$ using epsilon delta definition.
0
votes
2answers
36 views

Examples of Functions f and g such that lim f(x)g(x) exist but lim f(x) and lim g(x) doesnt

what such cases exist? Such that $\lim_{x\to a} f(x)g(x)$ exists even though neither $\lim_{x\to a} f(x)$ nor $\lim_{x\to a} g(x)$ exists.
0
votes
0answers
30 views

Strange question about magnetic dipole in a plane at infinite distance

Please allow me to ask something rather unusual and perhaps completely naive. Suppose I have an electric current in a circular loop in a plane. Consider it just in a mathematical sense. The loop has ...
2
votes
5answers
117 views

'Proof ' that $\ln(x)$ converges

Where is the flaw in the following 'proof '? $$\lim_{x \to \infty}\left[\frac{\mathrm{d}}{\mathrm{d}x}\left\{\ln(x)\right\}\right]=\lim_{x \to \infty}\left[\frac{1}{x}\right]=0 \implies\lim_{x \to ...
0
votes
1answer
23 views

Limit involving directional derivatives [on hold]

If $z = f(x, y)$ is differentiable at $\textbf{x}_0 = (x_0, y_0)$ Find $ \lim\limits_{\textbf{x} \to \textbf{x}_0}\dfrac{f(\textbf{x})-f(\textbf{x}_0)-\nabla f\left(\textbf{x}_0 \right)\cdot ...
4
votes
4answers
241 views

Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
1
vote
1answer
18 views

Limit involving sinus to show resonance-behavior

I got the following term: $$ - \frac{\omega}{\mu^2 - \omega^2} \frac{1}{\mu} \sin(\mu t) + \frac{1}{\mu^2 - \omega^2} \sin(\omega t),$$ with $t, \mu \in \mathbb{R}$ and $\mu > 0$ and i'm ...
2
votes
2answers
49 views

Help Evaluating $f'(-2)=\lim_{h\to0}\left(\frac{(-2+h)e^{-2+h}+2e^{-2}}{{h}}\right)$

Does anyone know how to evaluate the following limit? $$ f'(-2)=\lim_{h\to0}\frac{(-2+h)e^{-2+h}+2e^{-2}}{{h}} $$ What is $f(x)$ ? I want to see a step by step solution if possible.
0
votes
2answers
37 views

finding limit value of a given function-real analysis

what does $\lim_{n->\infty}$ $\frac{1} {\sqrt{n}}$ $(\frac{1} {\sqrt{1}+\sqrt{3}}$ + .....+$\frac{1} {\sqrt{2n-1}+\sqrt{2n+1}})$ equals? $\sqrt{2}$ $\frac{1}{\sqrt{2}}$ $\sqrt{2}+1$ $\frac{1} ...
1
vote
2answers
55 views

Legitimacy of a solution

Problem $$a_n=(1+\frac{1}{n^2})(1+\frac{2}{n^2})...(1+\frac{n}{n^2})$$ Find $\lim_{n\rightarrow\infty}a_n$ My solution We have ...
0
votes
1answer
20 views

Calculate derivative of multicase function involving exponentials as $x \to 0$ by definition

While this seemed (and probably does seem to some of you) like a simple question a first it stumbled me a bit. We were asked to calculate the derivative of: $$f(x) = \left\{ \begin{array}{lr} ...
0
votes
2answers
69 views

A limit of an Integral

Consider the following limit $$K=\lim_{x\rightarrow \infty}\frac{1}{x(1-x)}\left(1-\int_{\mathbb{R}}g(y;x)^x f(y)^{1-x}\mathrm{d}y\right)$$ where $f$ and $g$ are any continuous probability density ...
6
votes
2answers
57 views

Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$

Given a Heaviside function $$f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ Letting $a$ and $b$ be fixed constants, show that for all values of ...
1
vote
2answers
35 views

Limit of a quotient with a radical in the numerator

I have a limit but I'm so confused in how to rationalize the numerator because it has two numbers separated. How should I change the signs, please help me out. $$\lim \limits_{x \to ...
1
vote
3answers
58 views

Help Evaluating $\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\frac{\pi}{2}-x}-\tan {x}\right)$

Does anyone know how to evaluate the following limit? $$\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\frac{\pi}{2}-x}-\tan {x}\right)$$ The answer is 0 , but I want to see a step by step solution if ...
6
votes
4answers
72 views

Help Evaluating $\lim_{x\to0}(\frac{\sin{x}}{{x}})^{\frac{1}{x}}$

Does anyone know how to evaluate the following limit? $\lim_{x\to0}(\frac{\sin{x}}{{x}})^{\frac{1}{x}}$ The answer is 1 , but I want to see a step by step solution if possible.
0
votes
4answers
97 views

Help Evaluating $\lim_{x\to+\infty}\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$ [on hold]

Does anyone know how to evaluate the following limit? $$\lim_{x\to+\infty}\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$$ The answer is 1 , but I want to see a step by step solution if possible.
2
votes
2answers
48 views

Calculate the limit $\lim\limits_{x\to\infty} (a^x+b^x-c^x)^{\frac{1}{x}}$

Calculate the limit $\lim\limits_{x\to\infty} (a^x+b^x-c^x)^{\frac{1}{x}}$ where $a>b>c>0$. First, $$\exp\left( \lim\limits_{x\to\infty} \frac{\ln(a^x+b^x-c^x)}{x} \right)$$ Next, ...
-1
votes
0answers
35 views

For a map $f :\mathbb R^m\to\mathbb R^n$, prove that $\lim_{x\to a} f(x)=L$ if and only if $\lim_{x\to a}\|f(x) − L\|=0$

Could anyone help me with this proof? Given a map $f : \mathbb R^m \to\mathbb R^n,$ prove that $\lim_{x \rightarrow a} f(x) = L$ holds if and only if $$\lim_{x ...
0
votes
0answers
18 views

Is the following limit of an integral positive?

In one of my problems I need to show if the following holds $$0<\lim_{{g\rightarrow f},\,{\alpha\rightarrow\infty}}\int_\mathbb{R}\log^2(g/f)(g/f)^\alpha f \, \mathrm{d}\mu<\infty$$ Here $g$ ...
1
vote
2answers
42 views

How to solve these problems without using L'Hopital's Rule? [on hold]

$\lim _{x\rightarrow 0^{+}}\dfrac {\ln \left( \sin x\right) }{\ln \left( \tan x\right) }$ $\lim _{x\rightarrow 0}\left( \dfrac {\sin x}{x}\right) ^{\dfrac {1}{x}}$ $\lim _{x\rightarrow \dfrac {\pi ...
1
vote
1answer
56 views

Find the limit of $ f(x)=\begin{cases}x^2&\text{$x\in\mathbb{Q}$}\\0&\text{$x\notin\mathbb{Q}$}\\\end{cases}$ for $x\rightarrow0$ [on hold]

$ f(x)= \begin{cases} x^2 & \text{$x\in \mathbb{Q}$}\\ 0 & \text{$x\notin\mathbb{Q}$}\\ \end{cases} $ How do we prove that $\lim\limits_{x\to0}f(x)=0$? BTW, what do we call ...
3
votes
0answers
54 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
-4
votes
1answer
71 views

How prove this integral with the limit [on hold]

Find the limit $$\lim_{x\to 1^{-}}\left(2\int_{0}^{x}\dfrac{\ln{(1-t)}\ln^2{(1+t)}}{1-t}dt-2\ln{2}\ln{(1-x)} ...
3
votes
2answers
89 views

Behaviour of the function $\ln(1+ x^2)$

Thus function has derivative equal to: $\frac{2x}{1+x^2}$. This indicates that it will flatten out while approaching infinity, ie, should have an asymptote. Yet, the function does not have any real ...
1
vote
3answers
41 views

Why $\lim_{x\to0}\frac{\int_0^x\frac{t^2}{t^4+1}dt}{x^3}=\lim_{x\to0}\frac{\frac{x^2}{x^4+1}}{3x^2}$?

There is this limit question that I think we need to use the Fundamental Theorem of Calculus, but there is one little doubt during the process. ...
1
vote
2answers
50 views

Second derivative using limits

If f is a function that is two times differentiable at x = a then: $\lim\limits_{h \to 0} \frac{f(a+h)-f(a)-hf'(a)}{h^2/2}=f''(a)$ I don't know how to prove or disprove this. I know I have to use ...
5
votes
4answers
173 views

Solve a limit with radicals

I don't know how to solve this limit. What should I do? $$ \lim_{x\to 0} {\sqrt{x^3+2x+1}-\sqrt{x^2-3x+1} \over \sqrt{4x^2-3x+1} - \sqrt{2x^3+6x^2+5x+1}} $$ Thank you!!
0
votes
2answers
115 views

Any proof that verify why the limit of the difference is the difference of the limits?

I did a research on internet and books about why the difference of the limits is the difference of the limits, but i didn't get any result of this proof. I would appreciate if somebody can help me. ...
0
votes
2answers
131 views

Why the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 is 1? [duplicate]

I need a rigorous proof that verify why the limit of $\dfrac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$. I tried before but i do not know how start this proof. I would appreciate if somebody help me. ...
1
vote
1answer
34 views

Switching Limits and summation

I'm currently working on proving some theorems and there is one recurring problem that I somehow can't solve. $a_n$ is a real sequence in either $[0,1]$ or $\mathbb{R}$ that approaches $0$. ...
1
vote
3answers
70 views

verify $\lim_{x\rightarrow0}(4x^2+2x+5)=5$

Verify: $$\lim_{x\rightarrow0}(4x^2+2x+5)=5$$ On a simple linear function it's easy to use the limit definition "$|f(x)-L|$ becomes arbitrarily small" but it won't work in this situation. But I'm ...
3
votes
3answers
73 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...
2
votes
3answers
101 views

Find $\lim_{n\rightarrow \infty}\sqrt{2+\sqrt{2+\sqrt{…+\sqrt{2}}}}$ [duplicate]

This is a nice limit $$\lim_{n\rightarrow \infty}\underbrace{\sqrt{2+\sqrt{2+\sqrt{...+\sqrt{2}}}}}_{n\text{ times}}$$ and it is solved with well-known trigonometry formulas. The result is 2. The ...
1
vote
0answers
39 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...