Limits Definition and 1000 Threads

Homework Statement I'm wondering why we can't use less than or equal to for the formal definition of the limit of a function: Homework Equations lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)<δ implies abs(f(x) - L)<ε Why not: lim x→y f(x)=L iff For all ε>0 exists...

Homework Statement Prove the converse of the Cauchy Criterion for Limits. Let I be an interval that either contains the point c or has c as one of its endpoints and suppose that f is a function that is defined on I except possibly at the point c. Then the function f has limit at c iff for...

Suppose one needs to evaluate a definite integral over a singularity, like: -\int_{-1}^3 \frac{1}{x^2} dx The textbook way to do so is to split the integral into two parts around the singularity and take the limit, like so: \lim_{b\rightarrow 0} -\int_{-1}^b \frac{1}{x^2} dx and...

Homework Statement Let f(x,y) be defined : f(x,y) = 0 for all (x,y) unless x4 < y < x2 f(x,y) = 1 for all (x,y) where x4 < y < x2 Show that f(x,y) → 0 as (x,y) → 0 on any straight line through (0,0). Determine if lim f(x,y) exists as (x,y) → (0,0). Homework Equations Polar co - ordinates...

Homework Statement Evaluate lim h->0 (f(4+h)-f(4))/h given f(x)=|x-4|-4 2. The attempt at a solution Not too sure how to do this with absolute values, I've tried just subbing in small values for h, which gave me -1, but as this is an online question for marks, I am not too sure. Any...

Homework Statement Homework about a result in probability theory, but I don't understand one of the steps: Let f(x) be the PDF of a continuous R.V which takes only non-negative values. Why is the following true? \int^{\infty}_0\int^{\infty}_xf(t) \mathrm{d}t \mathrm{d}x =...

Homework Statement lim [(5^(n+1))+(7^(n+1))]/(5^n-7^n) as x→infinity lim [sec x - tan x] as x → ∏/2

Doesn't solving limits by substituting values defeat the point? For example: you can solve a limit of a quadratic by just subbing in a x value. But how do we know limit as x approaches a is the same as f(a)?

Homework Statement [SIZE="3"]\stackrel{lim}{n\rightarrow \infty} (-1)^n \frac{n}{n + 1} Homework Equations The Attempt at a Solution The answer is that the limit oscillates between -1 and 1, but I was wondering if there was an analytic was of showing this.

Okay so here is the deal, I understand the concept behind a limit just fine. However (for some reason) whenever I am asked to graph or visualize the concepts of these notations that my book is using I am failing. So that makes me think that maybe I do not understand limits... for instance my...

Does anyone know of websites where I can find many problems on the topic in the title line (my textbook has far too few)? Thanks!

Homework Statement Let f be the function such that f(x)=1+^2 if x is irrational and f(x)=1+4x^4 if x is rational. Use the Squeeze Theorem to find (lim x->0)f(x). Clearly, for all real numbers x, f(x)>=1. Next, we note that for a real number x, x^2−4x^4=(x^2)(1−4x^2)>=0 if and only if ? >=0...

Hi guys, I'm really new to calculus and limits and have been trying to have a good crack at the following question. Sorry if I haven't written the problem out in the most acceptable format. lim (9-3√x)/(9-x) x→9 Substituting 9 gives you 0/0 and indeterminate. I tried multiplying the...

Homework Statement (Squeeze Principle) Suppose f, g, and h are real-valued functions on a neighborhood of a (perhaps not including the point a itself). Suppose f(x) ≤ g(x) ≤ h(x) for all x and limx→a f(x) = l = limx→a h(x). Prove that limx→a g(x) = l. (Hint: Given ε > 0, show that there is δ...

Homework Statement \lim_{a→0+}\frac{1}{a}- \lim_{b→0-}\frac{1}{b} Homework Equations The Attempt at a Solution I assume I would have to use the Cauchy definition of a limit to solve this, but I was wondering if there were any alternative ways. BiP

Prove that \lim_{a→0-}\frac{1}{a} = \lim_{b→0+}\frac{1}{b} Is this statement true? How can one prove its truth/falsity? Would we need to use the precise Cauchy definition of the limit to do this? BiP

Can somebody tell me how to find one sided limits for greatest integer function, say Lim x->n-, [[x/2]]. that is limit x approaches n from left of [[x/2]] where [[]] represent greatest integer function and n is any integer. I know how to find one sided limits for simple [[x]].

Homework Statement Homework Equations Only ones I know that are relevant are the equations shown in the question (for the left and right limits). The Attempt at a Solution The difficulty that I am having right not is comprehending what is really being asked from the question. From...

1. Evaluate 2. lim χ→0 [((χ+1)^1/3) -1] / χ 3. The hint is [a^2 + b^2 = (a+b)(a^2 + ab + b^2)]. But i don't get it how can that help me solve this because it is a cube root and not to the power of 3. Without thinking about the hint, I have attempted it and i think it is undefined for the...

suppose there is a function f(x), and it's limit as x goes to infinity is 0. Is there a theorem that says it's derivative, f'(x), also approaches 0 as x goes to infinity? Thanks.

Homework Statement \lim_{x\rightarrow \frac{\pi}{2}} \frac{tan(2x)}{x - \frac{\pi}{2}} Homework Equations The Attempt at a Solution I was given a couple of hints: use substitution, and that there isn't any need for the tangent double angle formula. I would have never thought to use...

Limit of otherwise equal summations is uneuqal due to "indeterminate difference"? Hello! I was working through a section on summation in a maths textbook because of a recent question I asked on the forum, and I came across something which is confusing. I am not sure if it is necessary, but...

Hello, I am going through Whittaker's treatise on Classical Mechanics. In chapter 3 he derives the equation of motion for a simple pendulum, and I have a question about his method. Starting from the general form for the equation of energy (s is the path): [SIZE="5"]\frac{m}{2}\dot{s}^2...

Homework Statement #1. If limit[x->a]f(x) exists, but limit[x->a]g(x) doesnt, limit[x->a](f(x)+g(x)) doesn't exist. T/F? (Proof or example please) #2. prove that if f is continuous, then so is |f| #3. f(x) = [[x]]+[[-x]] for what a does limit[x->a]f(x) exist? Where is f discontinuous...

Hi All, I need your help, we we have an intergral like this ∫^{T}_{max \in \{0, t\}} f(x) dx what is the meaning of the lower limit in this integral? Thanks in advance

Homework Statement lim(x+3)=? x→3 Homework Equations Substituting x=3, I think. The Attempt at a Solution lim(x+3)= 3+3= 6 x→3 So we have got this straight lined graph, f(x)=x+3, and the answer I got shows... what exactly?

we set limits by what we can measure; therefore, it is what we have. This is my general understanding. I think it extends at both ends and we haven't found the means to measure it as of current. what are we doing to expand the spectrum? my simple question.

A limit is the value that a function approaches (without necessarily being equal to) as x approaches a specific value. A limit can only exist if the limits approaching from the left and the right both exist and are equal. the analogy I've been going off is the idea of a force field or a...

Homework Statement The tangent to the function y=3x(x-3) at point P(2,-6) is the hypotenuse of a right triangle that forms with the coordinate axes. Find Area The Attempt at a Solution First of all, i know that i A=BxH/2 so i need the opposite and adjacent sides of this triangle...

Homework Statement There isn't a problem statement, I'm just confused about something. Homework Equations The Attempt at a Solution Ok, for a while I didn't understand how cancelling in the following limit works: (x+5)(x-5) lim ---------- x--> 5...

Homework Statement lim x-->0 sin(pi/x) sqrt(x^3+x^2) The Attempt at a Solution I was having trouble evaluating the above limit. Do I start by isolating x? For some reason, when it comes to trig functions such as this, I'm not sure how to simplify it. Also, what material would I have to...

Homework Statement I have attached the question, along with the solution in the picture attached. This is one of the few questions I have encountered that I completely have no idea what the solution is trying to do... It's like they do not make any sense at all! Confused by 1. The...

Homework Statement Prove that lim[x->5] 3x^2 - 1 = 74 Homework Equations Epsilon-delta stuff. The Attempt at a Solution Ok, so first we have to find delta in terms of epsilon, and this is particularly sucky because it's a nonlinear function. Here's my attempt: |3x^2 - 1 - 74| =...

Hello. I use a Fifo IC AL422B with a 1 MHz write clock and a 625 kHz read clock. I just discovered that 1 MHz is the low speed limit for the clocks. I can speed up the write clock to 2 MHz but unfortunately I can't change read clock speed. As these clocks ( or at least one of them) are used for...

Whilst messing around with some geometry pertaining to the n-sided regular polygon, I stumbled upon this equation which I could not find anywhere on the internet. \pi = \lim_{n \to \infty} n \sin \frac{\pi}{n} But if we take this to be true then, by substitution, this is also true: \pi =...

Hi people.. Snell's law reads sinθ = (v2/v1)sinβ Suppose that v2/v1 > 1, then we can make sinβ as close to 1 as we like, even close enough to make sinθ>1 as Snell's law states. So what's wrong?

If I do this: \lim_{\alpha\rightarrow 0} \frac{sin\alpha}{\frac{2\alpha}{5}} = \lim_{\alpha\rightarrow 0} \frac{5sin\alpha}{2\alpha} = \lim_{\alpha\rightarrow 0} \frac{5}{2}\cdot \frac{sin\alpha}{\alpha} Am I allowed to do this? \frac{5}{2} \cdot \lim_{\alpha\rightarrow 0}...

Hello! Do you know, or know where to find, the limits of imaging via waves, ie. let's say we have a biphase material, can we identify precisely the margin between phase? Or can we detect any kind of material? Thanks

1. Determine whether the sequence converges, and if so find it's limit. {\frac{(n + 1)(n + 2)}{2n2}} 2. The attempt at a solution lim n→∞ \frac{(n + 1)(n + 2)}{2n2} = lim n →∞ \frac{(n2+ 3n + 2)}{2n2} = lim n→∞ \frac{n2}{2n2} + \frac{3n}{2n2} + \frac{1}{n2} = lim n→∞...

I am aware that, in statistics, things get difficult as soon as they get nonlinear. And taking the reciprocal of a quantity is a nonlinear operation. I have some data that would form a nice looking straight line, except for random error scattering it around the line. I have a total of about...

Suppose that (X,Y) is uniformly distributed over the regiondefined by 0≤ y ≤ 1-x2 and -1≤ x ≤ 1. a) find the marginal densities of X and Y Attempted solution: So first I have to find the joint density function which ends up being fxy(x,y) = 3/4 and then from that I would solve...

Hi guys, I just started reading an introductory book on analysis. I'm up to the part where they talk about functions now, and I'm getting lost. The theorem that I'm having trouble envisioning is: Let f: D-> R and let c be an accumulation point of D. Then limx->cf(x)=L iff for each...

I am currently reading about finding areas under graphs using summations, specifically taking the of the number of rectangles, n, goes to infinity. My books says that "because the same limit value is attained for both minimum value f(mi) and the maximum value f(Mi), it follows from the squeeze...

I have grasped everything in the first 2 segments of my limits chapter, but somehow I am missing out on this problem. lim (f(x)-4)/(x-2) = 1...find lim x->4 x->4 f(x) I am missing something very fundamental and can not find an example...

Homework Statement Problem reads: find inverse Laplace transform of f(t) of F(s)=(2s+3)/(s(s2+7s+10) What is the value of the function f(t) at t=0 and t=∞?Homework Equations Inverse laplace transformThe Attempt at a Solution I solved F(t) down to F(t)= .3/s+0.166/s+2-0.465/s+5 thus lead me to...

Homework Statement Find the limits of the following Homework Equations We can't use L'Hôpital's rule as it's not in our syllabus.We have to use only The Attempt at a Solution I attempted a lot but can't find a way to solve these three (out of about 80 limits) pls show me the...

Homework Statement Calculate the integral: ∫ ∫_R (x+2y)^{-(1/2)} dxdy Where R is defined as points (x,y) which satisfy: x-2y ≤ 1 and x ≥ y2 + 1 Homework Equations So basically I'm completely stuck on this exerzice. As far as I can see, you could make the x limit go from...

Hi everybody, I am trying to solve the following problem and I get stuck on the last question. I would appreciate a lot that someone helps me . Here is the problem: Let D be the region bounded from below by the cone z= the root of (x^2 + z^2), and from above by the paraboloid z = 2 – x^2 –...

Homework Statement A question asks to calculate the integral over the region R given by: x^2 + y^2 <= 4 0 <= y <= 2 Which would be the upper half of a circle of radius 2 centred on the origin. The integral is done in the book I have and the limits of x are given as -2 to 2, which I...

Homework Statement I need to evaulate ∫ ∫S dS where S is the surface z = x² + y², 0 ≤ z ≤ 4. Homework Equations dS = √( 1 + ƒ²x + ƒ²y)dxdyThe Attempt at a Solution dS = √( 1 + 4x² + 4y²)dxdy here's the problem what are the limits to the surface integral? no clue.. dx means i should find...