Limit Definition and 999 Threads

Find the limit of 5/(x^2 - 4) as x tends to 2 from the right side. Approaching 2 from the right means that the values of x must be slightly larger than 2. I created a table for x and f(x). x...2.1...2.01...2.001 f(x)...12...124.68...1249.68 I can see that f(x) is getting larger and larger...

Find the limit of (3x)/(x - 2) as x tends to 2 from the left side. Approaching 2 from the left means that the values of x must be slightly less than 2. I created a table for x and f(x). x...0...0.5...1...1.5 f(x)...0...-1...-3...-9 I can see that f(x) is getting smaller and smaller and...

Find the limit of x/(x^2 - 4) as x tends to 2 from the right. If I plug x = 0, I will get 0/-4 = asymptote. Again, is graphing the best to do this one? I can also create a number line. <----------(-2)----------(0)---------------(2)--------> I can then select values for x from each interval...

Find the limit of (2x + 1)/(x + 4) as x tends to - 4 from the right side. I know there's a vertical asymptote at x = -4. I think the best way to solve this problem is by graphing the function. I am not too sure about how to solve algebraically. I am thinking about the number line...

Greetings! In statistical mechanics, when studying diffusion processes, one often finds the following reasoning: Suppose there is a strictly positive differentiable function ##f: \mathbb{R} \rightarrow \mathbb{R}## with ## \lim_{x \rightarrow +\infty} {f'(x)} = a > 0##. Then for sufficiently...

Hello. Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist. But can a quotient of the two acutally approach a certain value? lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...

I tried taking e^ln but to no avail. Please help! Thanks. My attempt: $$\lim_{x\to 0^+}(1-\cos (\sqrt x))^{\sin(x)}$$ $$\lim_{x\to 0^+}e^{\ln (1-\cos\sqrt x)^{\sin x}}$$ $$\lim_{x\to 0^+}e^{{\sin x}\ln (1-\cos\sqrt x)}$$ $$\lim_{x\to 0^+}\exp(\frac{\ln (1-\cos\sqrt x)}{1/\sin x})$$ If I apply...

The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential...

I wonder if the limit of the following can be converted into integral or some elegant form as N tends to infinity: \[ \sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right) \] If we plot or evaluate the value then it does appear that the series converges very fast...

Hi all, I'm a little confused about something. Force-extension graphs and stress-strain graphs are always both straight lines up until the limit of proportionality, implying both the spring constant and the Young modulus are constant up until then. For a force-extension graph, Hooke's Law...

Can someone please tell me how to solve a limit problem like this? $$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$ This is my attempt to solve the problem: $$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$ $$= \lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} -...

Hi guys, I am having difficulties in solving this limit. Below, I'll attach my procedure which ends up in the indeterminate form ##0\cdot \infty##... How could I solve it? $$\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x) \rightarrow \lim_{x \to +\infty}(x\sqrt[3]{1-\frac{4}{x}}-x) \rightarrow...

Large wind turbines have become very efficient and have a power coefficient close to that defined by Betz. However, large wind turbines are stopped when the wind is too strong, not because they produce too much, but because their blades are subject to bending stresses which may break their...

I have to prove that \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]= a (in order to show that the indeterminate form of the type 0^0 can be any positive real number). This is what I did: Let y = \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]...

Hi, PF This is the quote: "If ##m## is an integer and ##n## is a positive integer, then 6. Limit of a power: ## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##" What do I understand? -whenever ##L>0## if ##n## is even: ##m##...

I'll write my considerations which lead me to get stuck on the ##\infty-\infty## form. $$\lim_{x \to +\infty }\sqrt{x^{2}-2x}-x+1 \rightarrow |x|\sqrt{1-0}-x+1$$ And I have no idea on how to go on...

For example: A person who gains degrees in 5 different fields. Will this person be able to refer to information learned in all 5 fields and bring it up at will? Going by my own understanding of my self it seems to fade away when something new is learned and focused on. But once you revise over...

Here's my attempt: I would like to confirm my answers.

I have already in a previous task shown that A is not irreducible and not regular, which I think is correct. I don't know if I can use that fact here in some way. I guess one way of solving this problem could be to find all eigenvalues, eigenvectors and diagonalize but that is a lot of work and...

How could I show that this limit: ##\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}## is equal to 0? In the expression above ##T_{4N}## is the Chebyshev polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...

Summary:: x Let ## \{ a_{n} \} ## be a sequence. Prove: If for all ## N \in { \bf{N} } ## there exists ## n> N ## such that ## a_{n} \leq L ## , then there exists a subsequence ## \{ a_{n_{k}} \} ## such that ## a_{n_{k}} \leq L ## My attempt: Suppose that for all ## N \in {\bf{N}} ##...

A few days ago, I noticed that there is a Linux terminal command 'tc', which can be used for limiting the download and especially upload speeds of your internet connection. When I write sudo tc qdisc add dev eth0 root tbf rate 100kbit latency 10ms burst 1540 in the terminal, the download speed...

In a statistical mechanics book, I learned about the degenerate pressure of a Fermi gas under the non-relativistic regime. By studying the low-temperature limit (T=0), we got degenerate pressure is ##\propto n^{5/3}## (n is the density). And then I was told that in astrophysical objects, the...

If f(a) = 2, f'(a) = 1, g(a) = –1, and g'(a) = 2, the value of $$\lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}$$ is ... A. 1 B. 3 C. 5 D. 7 E. 9 $$\lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}=\lim_{x\to a}\frac{2g(x)+f(x)}{x-a}$$. How to determine the f(x) and g(x)? And when...

$$\lim_{x\to0}\frac{sin2x+sin6x+sin10x-sin18x}{3sinx-sin3x=}$$ A. 0 B. 45 C. 54 D. 192 E. 212 Either substituting or using L'Hopital gives $$\frac00$$. Is there any way to simplify it and make the result a real number?

Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question. Question: Let P(k,k^2) be a point on the parabola y=x^2 with k>0. Let O denote the origin. Let A(0, a)denote...

I know it's probably an easy one, but I'm getting confused on how to treat that exponential numerator in order to escape from the indeterminate form ##\frac{0}{0}##

I'm trying to determine why $$ \lim_{N \rightarrow +\infty} ln( \frac {N!} {(N-n)! N^n}) = 0$$ N and n are both positive integers, and n is smaller than N. I want to use Stirling's, which becomes exact as N->inf: $$ ln(N!) \approx Nln(N)-N $$ And take it term by term: $$ \lim_{N...

I can not follow this mathematically. I am guessing one of the signs is incorrect? Can anyone verify ?

Given is the following: lim x-2 of f(x)=2 prove (using delta, epsilon definition of a limit) that a delta exists so that when [x]<delta then f(x)>1 I came up with when [x-a]<delta (f(a)-epsilon<f(x)< f(a) + epsilon) so f(a)-epsilon>1 so epsilon<f(a) -1 but I don't know how to prove this or...

How would one prove that, for any integer x (where x>1), this series always reduces to 2? if is_even(x) x=x+(x/2) else x=(x+1)-((x+1)/2) end

Not sure how to go about this. Would relying on a hole or asymptote work?

(a) I thought perhaps a parameterization would be the place to begin given all the squared terms. x=rcos(u)sin(v) y=rsin(u)sin(v) z=rcos(v) That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k)) Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k)) I'm not sure how...

Attached is a proof that $$ lim \inf \subset lim \sup $$ for an infinite sequence of non-empty sets. The basic idea is to use the axiom of choice/well ordering theorem to show that 1) there is something in the limit superior 2) there is something in the limit superior that is not in the limit...

If a voltage source is sinusoidal, then we can introduce a phasor map and come up with equations like$$V_0 e^{i \omega t} = I(R + i\omega L + \frac{1}{\omega C} i)$$where ##I## would also differ from ##V## by a complex phase. But if you set ##\omega = 0##, which would appear to correspond to...

The apparent length of a rod is determined to be$$\tilde{L}(x_0) = \gamma L + \beta \gamma \sqrt{D^2 + (\gamma x_0 - \frac{L}{2})^2} - \beta \gamma \sqrt{D^2 + (\gamma x_0 + \frac{L}{2})^2}$$I am trying to determine expressions for ##\tilde{L}(x_0)## when ##x_0 \rightarrow -\infty## and ##x_0...

The Schwarzschild metric seems to model, for example, the earth’s gravity field above the earth’s surface pretty well, even though the Earth is not really a golf-ball sized black hole down at the center. Can the same be said for the Kerr metric? Does it model a rotating extended body’s gravity...

All the references I find refer to safely charging lithium cells by a method like this: https://www.powerstream.com/li.htm The next page shows the effects on capacity of charging to less than the 4.2 V terminal cell voltage. For example, charging to 4.0 V still provides 73% of the capacity...

Hello everyone, I need to find this limit . What I tried is that , but clearly, 1/x diverges so I don't think it was very helpful. Could someone help me what I need to do please?

I need to solve this limit without L'Hôpital's rule. Could someone give me a hint what I need to do please? I just can't find this algebraic trick. Thank you in advance!

We have the limit of the sequence ##\frac{a^n}{n}## where ##a>1##. I know it is ##+\infty## and i can prove it by switching to the function ##\frac{a^x}{x}## and using L'Hopital. But how do i prove it using more basic calculus, without the knowledge of functions and derivatives and L'Hopital...

(1 + a^x)^(a^(-x)) Let's assume a, say, two. the limit of it, with x tending to infinity, is one, but i was thinking... Calling 2^x by a, we have that when x tend to infinity, so do a, So: that is euler number no? Contradictory... where am i wrong?

$$ \lim_{x \to 0} [ \frac{J_{p}(x)}{Y_{p}(x)} ] $$

Hello everybody, could you help me with this problem please? I have to find a derivative in x0 of this function (without using L'Hospital's rule): I used the definition , but I don't know what to do next. Thank you.

If there is no ##(-1)^2## factor, I can find the limit. But, now I have no idea how to find limit for the ##(-1)^\infty##. I thought ##(-1)^\infty## is an indeterminate form. So, how to modify this? Thanks!

I tried to substitution n = infinity so I got (infinity)*sin (1/infinity). I thought 1/infinity is approaching zero. So, sin (1/infinity) is same with sin (0). With these idea, my solution is lim n--> infinity n sin(1/n) = 0. But, the answer book say that the answer is 1. I tried another...

First I assume that $$(1+n^2)^\frac{1}{\ln n}=\exp {\ln (1+n^2)^\frac{1}{\ln n}} $$But, $${\ln (1+n^2)^\frac{1}{\ln n}}={\frac{\ln (1+n^2)}{\ln n}}$$ By L'Hopital Rule, I got $$\lim_{n\to\infty} {\frac{\ln (1+n^2)}{\ln n}}=\frac{\lim_{n\to\infty} (\frac{2n}{1+n^2})}{\lim_{n\to\infty} 1/n}$$...

Ok all I did was DesmosNot real sure how to take limit

Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.