Limits Definition and 1000 Threads

I'm retaking Calculus I right now during the summer. It's been quite a while but I am pretty much breezing through it except for the limit section. I remember the first time around, my prof didn't put ANY emphasis on limits and I haven't really had to do much of anything with limits since then...

Homework Statement \stackrel{lim}{x→1}(\frac{p}{1-x^p}-\frac{q}{1-x^q}) Homework Equations The Attempt at a Solution I tried writing 1-xp as (1-x)(1+x2+x3...xp-1) and same with 1-xq but i don't seem to find any way further.

Really couldn't catch the concept on epsilon and delta in limits. Let ∂x=x2 - x1 In finding a gradient the value ∂y is taken at certain value. But in finding area using integral, the ∂y is seen to taken as zero. F(x2)=F(x1) Maybe one multiplication and the other is division.

limits -- Prove that xsin1/x approaches 0 near 0 Prove that xsin1/x approaches 0 near 0. Similiar Proof from book |sin1/x| ≤ 1 | xsin1/x | ≤ |x| for all x not equal to 0, so we can make |xsin1/x|< ε by requiring that |x| < ε and not equal to 0. MY QUESTION: Prove x2sin1/x...

I have attached this definition that my book provides. My question is does that part "for each M > 0 there exists δ > 0 such that f(x) > M, mean that whenever you M close to the limit, you can find a δ that will give M1 that is closer to the limit?

Homework Statement The Attempt at a Solution I don't see how the above step is legal. I plugged in 4 for z and 9 for z and it works, but what is this move called? it's neither rationalizing the numerator nor rationalizing the denominator.

I am studying unit on limits and one of the example given to prove and established limit simply doesn't make sense. in the given solution of the example righhand side is simply not making sense to me - please see attached document and anyone can throw some light on this will be great so i can...

Homework Statement Determine if the following series converges or diverges and find the limits for those series that converge. (a)\;\sum^{\infty}_{n=0}\frac{1}{3^{n-1}} (b)\;\sum^{\infty}_{n=0}\frac{4n^2+n}{(n^7-n^3)^{1/3}} (c)\;\sum^{\infty}_{n=0}\frac{\sin^2n}{2^n} Homework Equations...

Homework Statement For example in: lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)] This limit does not exist (according to textbook), but if you use squeeze theorem since y^2<(x^2+y^4) y^2/(x^2+y^4) <= 1 and therefore 0 <= (xy^2)/(x^2+y^4) <= x as x--> 0 so lim (x,y) -> (0,0)...

Homework Statement For the double integral [SIZE="3"][SIZE="4"]∫[0,1]∫[0,x^3] e^(y/x) dxdy (a) sketch the region of integration (b) evaluate the integral and (c) re-express the integral with the order of integration reversedHomework Equations NoneThe Attempt at a Solution The problem is that...

Help choose the limits of the following volume integrals: 1) V is the region bounded by the planes x=0,y=0,z=2 and the surface z=x^2 + y^2 lying the positive quadrant. I need the limits in terms of x first, then y then z AND z first, then y and then x. And also polar coordinates, x=rcost...

Homework Statement Assume that f is a monotone increasing function defined on \mathbb{R} and that for some x_0\in \mathbb{R} the left and right limit coincide. Can you prove that f is continuous at x_0? Either give a complete proof or a counterexample. Homework Equations The...

what rule are you supposed to follow when you evaluate a rational function at 0? eg in this problem if you evaluate at s=0 for the one under "result" it will be different from the value obtained for the one under "alternate forms"...

Homework Statement 2. Show that the function is continuous on the given interval. (a)f(x)= (2x+3)/(x-2) range:(2, infinity) (b)f(x) = 1- sqrt(1-x^2) range:[-1,1] 3. Prove that the following limits do not exist. (a) lim x tends to 0 ( absolute|x|/x) (b) lim x tends to 3 (2x/(x-3))...

I would imagine that an alternating series that goes of to infinity doesn't have a limit because it keeps switching back and forth, but I can't find anything in my textbook about it. I just want to make sure that this is right.

Hi all, I have a little out of track question and I was forced to consider this after reading FQXI Essay competition title Is Reality Digital or Analogue and Kant's Critique of Pure reason simultaneously. If I am not wrong, according to Kant, there are limits of pure reason. Is not the...

Can MMA take limits as 2 variables approach a value? I want to take the limit as h and k both approach 0 for f(x+h,y+k)-... Is there some trick to doing this?

I've been using the formula lim x-->a f(x)-f(a) -------- x-a I haven't had any problems until I was asked to find the slope of the tangent at the general point whose x-coordinate is a. How do I do it with an a instead of a point? I'm trying to find it for...

I don't know how to properly present my answer to find the limit of a converging sequence like (1/2)^n. I would just write something like this... y=1/x+1, x=∞ } y=1/∞+1=0+1=1 but the syllabus gives something completely different and my textbooks don't seem to cover this portion of the...

Let R be bounded by y=0, x=2 and y=x^2. Then ∫∫6xydA= ? (Note the integral is to be evaluated over R) Now what will be the lower limit of x. I took it to be 0 and the answer was 32. which turned out to be correct. Is their any way in such questions by which we can determine the lower limit...

Suppose that X is a random variable distributed in the interval [a;b] with pdf f(x) and cdf F(x). Clearly, F(b)=1. I only observe X for values that are bigger than y. I know that E(X|X>y)=\frac{\int_y^b xf(x)dx}{1-F(y)}. Moreover, \frac{∂E(X|X>y)}{∂y}=\frac{f(y)}{1-F(y)}[E(X|X>y)-y] I...

Homework Statement Is there such a number b such that lim x->-2 (3x^2+bx+b+3)/(x^2+x-2) exists? If so, find b and the limit.Homework Equations lim x->-2 (3x^2+bx+b+3)/(x^2+x-2)The Attempt at a Solution for the denominator we have zeroes at x = 1 and -2. so we need to get rid of the -2 part...

Consider 3 series: A(0) = 0, A(1) = 4; A(n) = 6*A(n-1) - A(n-2) + 4; B(0)=1, B(1) = 3, B(n) = 6*B(n-1)-B(n-2) - 4; and C(0) = -1, C(1) = -11, C(n) = 6*C(n-1)- C(n-2) -4. Is there a way to prove that the limit as n => infinity of A(n)/B(n) = -C(n)/A(n)? Note that series C is actually...

Homework Statement We're given a piecewise function g(x) = { x if x < 1 3 if x = 1 2-x^2 if 1<x<=2 x-3 if x > 2 ] and were asked: lim x-> 2- Homework Equations 2-x^2 if 1<x<=2 x-3 if x > 2 The Attempt at a Solution when i drew it out i was getting that the answer...

Homework Statement lim x->pi- cot(x) Homework Equations cot(x) = cos(x)/sin(x) The Attempt at a Solution so substituting pi into: cot(pi) = cos(pi)/sin(pi) = -1/0 so you have a negative over 0, approaching from the -ve side of pi wouldn't it be +infinity? why is it -infinity...

Homework Statement Let f be a continuous function on [1,∞) such that \lim_{x\rightarrow ∞}f(x)=α. Show that if the integral \int^{∞}_{1} f(x)dx converges, then α must be 0. Homework Equations Definition of an Improper Integral Let f be a continuous function on an interval [a,∞). then we...

Alright. I completely confused about determining the area between regions of polar curves. However, I do feel that I have a solid grasp in finding areas for single functions. For a given function in polar form, I know that I find the limits of integration by setting the function equal to zero...

Homework Statement Differentiate sin(ax), cos(ax) and tan(ax) from first principles. Homework Equations The Attempt at a Solution I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the...

Homework Statement Use the fact that a_n=a+(a_n-a) and b_n=b+(b_n-b) to establish the equality (a_n)(b_n)-ab=(a_n-a)(b_n)+b(a_n-a)+a(b_n-b) Then use this equality to give a different proof of part (d) of theorem 2.7. Homework Equations The theorem it is citing is: The sequence...

Homework Statement The Attempt at a Solution So I know that the limit as n → ∞ of (1 - \frac{1}{n})^n = \frac{1}{e}. Using this information, is it legitimate to observe: The limit as n → ∞ of (1 - \frac{1}{n})^{n ln(2)} = the limit as n → ∞ of ((1 - \frac{1}{n})^n)^{ln(2)} = e^{-1...

How can I use the directional derivative of a two variable function to show that the limit does not exist? For example, suppose I have a function f(x,y)=g(x)/f(y) and g(a)=f(b)=0 and the limit as x and y go to a and b is 0. How would I use the directional derivative to show that the limit at...

Hi, I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion. Here's what I have been trying to get at: \lim_{z\to z_o} z^2+c = {z_o}^2 +c |z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta...

I have been given a question to sketch the curve of y=sin(x). I have looked into finding the domain which I understand but I don't understand how I prove the x intercepts mathematically as when I make x=0 I obviously get a 0 value for y but a sin curve obviously intercepts and pi and 2pi etc...

Homework Statement the ques says: lim x tending to 0 [f(x)g(x)] exists. Then both lim x tending to 0 f(x) AND lim x tending to 0 also exist. True or False Homework Equations The Attempt at a Solution lim f(x)g(x) =lim f(x) * lim g(x) so if LHS exists then limf(x) and lim g(x) must exist so it...

Homework Statement Determine if limit exists: ## \left( !x_{,y}^{im}\right) \rightarrow \left( 1,2\right) =\dfrac {xy-2x-y+2} {x^{2}-2x+y^{2}-4y+5} ## Above is just lim (x,y)-->(1,2) Homework Equations The Attempt at a Solution ## c_{y}x=a,y\rightarrow b ## ## \lim _{\left( 1,y\right)...

Homework Statement Its just a general query about problems along these lines... f(x)=|x^2+3x-18|/(x-3) and a =3, discuss the limiting behaviour of f(x) as x→a^+, as x→a^- and as x→a. Homework Equations The Attempt at a Solution So my basic solution to these types of...

I know you can determine what number the limit of a series approaches, such as 2. Is there a way to do that in reverse? Is there a method where I can I come up with an infinite series that approaches a limit of, say, 7.5? or 11.75? And is the possible different series that will approach, say...

This is not homework. Earlier today I was trying to prove that if a limit of a certain function exists, then it's unique: limf(x)=a \wedge limf(x)=b (as x→x0) then a=b I began to use the sum of limits like so: limf(x)+limf(x)=a+a → lim2f(x)=2a (as x→x0) And the same thing for...

I know for multivariable function it doesn’t work. However if the function is in the form of f(z) and z is the only variables shown. Can I use it? Thanks

Homework Statement I'm trying to determine the limits for a double integral over a symmetric trapezoid or equilateral triangle. I'm not trying to determine the area, and therefore using symmetry to simplify the integration is not an option. The limits for the integration over the y-axis are...

Homework Statement Hi everyone! I'm pretty good with multivariable limits, but this one has me stumped: Find the limit or show that it does not exist: \underset{\left(x,y,z\right)\rightarrow\left(1,-1,1\right)}{\lim}\frac{yz+xz+xy}{1+xyz} Homework Equations The Attempt at a Solution I could...

Homework Statement I'm supposed to answer true or false on whether or not the sequence ((-1)^n * n) tends toward both ±∞ Homework Equations Uniqueness of Limits The Attempt at a Solution I did prove it another way, but I would think that uniqueness of limits (as a definition...

Homework Statement \lim_{x\rightarrow3} \stackrel{(t+1)^2}{(t^2+1)} Homework Equations The Attempt at a Solution if the bottom was t^2 - 1 i could factorise and cancel but when its t^2+1 I am not sure how to go...

Homework Statement show that if F:(a,∞) -->R is such that lim xF(x) = L, x --> ∞, where L is in R, then lim F(x) = 0, x --> ∞. Homework Equations The Attempt at a Solution Let F:(a,∞) →R is such that lim xF(x) = L, x → infinity, where L is in R. Then there exists an α> 0 where given ε...

Homework Statement (Sorry, having problems with math symbols) lim f(x,y) [(x^4)y]/(x^8+y^4) (x,y)→(0,0)Homework Equations Compare limits when a.) y=x b.) y=x^4The Attempt at a Solution The solutions are a.) limit approaches 0 b.) limit approaches 1 I think I understand in...

Homework Statement I'm having trouble calculating these limits. Homework Equations none The Attempt at a Solution 1. lim...((lnx)^5)/(x)= x->infinity (((lnx)^5)/x)/(x/x)= (((lnx)^5)/x)/1= How do I calculate from here? 2.lim ...(14lnx^2)/(6lnx^3)=...

I know that the solution is correct. But i do not know if i resolved it in the correct way. The images are attached.

Homework Statement Can someone tell if if these look right? Homework Equations none The Attempt at a Solution 1. lim (lnx)^5/x = x->infinity 5lnx/x = (5lnx/x)/(x/x)= (5lnx/x)/1 = 0/1 = 0 2. lim sinx/x^2= x->infinity (sinx/x^2)/(x^2/x^2)=...

Hi to all. How would I go about calculating the fatigue limit of leaf springs? Is there an equation out there in the form of N, cycles to failure= f(variables, material properties) Or will I have to use the Stress equations for leaf springs and use that stress value and then...

Please could someone assist me with this question Compute the limit of [FONT=CMR10]lim t→∞ [FONT=CMMI10]p(t) where p(t) = M*exp(D*exp(c*t))