1. ### Values of d where limit exists. lim x->d

Homework Statement For which value of d does the following limit exist? lim x->d ln [ (x2-13x+30) / (x-d) ] Homework Equations None The Attempt at a Solution I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...
2. ### Limit question

Homework Statement can you cancel the x and the |x|[/B] lim x→0 ( x(1-(cos(x))/|x| ) 2. Homework Equations The Attempt at a Solution At first i thought the limit would just be undefined as x approaches 0 but the answer to the problem is actually 0, so can you just cancel the x with the...
3. ### Sterling approximation of Beta Function

Homework Statement Homework Equations The Attempt at a Solution I think this problem is probably a lot simpler than I am making it out to be. However, when I apply sterling's approx., I get a very nasty equation that does not simplify easily. One of the biggest problems I have though is...
4. ### B Does 0.999... equal 1?

Does 0.999... equals 1? I know that this is a very basic well known concept but recently I stumbled across a video on Youtube in which the creator argues that the two are not equivalent I posted a comment arguing that in the case of Infinite sum of Σn=0 9(1/10)n you can find the sum of the...
5. ### Limit at infinity with radicals

Homework Statement lim as x tends to -∞ (x)^3/5 - (x)^1/5 Homework Equations The Attempt at a Solution The first thing I did was convert it into a radical so it becomes fifthroot√x^3 - fifthroot√x. Then I rationalized to get ( x^3-x)/(fifthrt√x^3+fifthroot√x) . I then divided the top by...
6. ### Limit of x as it approaches a variable

1. Problem statment Q. Use limit to find the instantaneous velocity at time t if the postition is p(t) at time t. p(t) = t + (1/t) at x = t 2. Homework Equations (Sorry dashes are in there to keep everything where it should be. I don't know how to make fractions in this and it was...
7. ### B How do you find the limit of this?

Hi! First time poster, I'm about to enter first year calc and thought that I could get ahead of the curve by checking out some questions beforehand. This showed up on one of the university calculus exams but I couldn't figure out how to do it. I tried to finding a common denominator but then was...
8. ### I Multivariable limits

I have found that multivariable limits are harder to find and/or prove that something exists. Do you have any recommendations, given questions like "find(if exists) the limit....". For example, I have no idea how to even start thinking about the following limit(if it exists or not, and if it...
9. ### Limits - composed functions

The problem $$\lim_{x \rightarrow \infty} \frac{(\ln x)^{300}}{x}$$ The attempt ## \lim_{x \rightarrow \infty} (\ln x)^{300} = \infty## since ## \lim_{x \rightarrow \infty} f(x) = A## and ## \lim_{x \rightarrow \infty} g(x) = \infty ## thus ## \lim_{x \rightarrow \infty}f(g(x)) = A ##. ##...
10. ### Limit fraction

The problem \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} The attempt \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} = \lim_{x\rightarrow \infty} \frac{x^4(1 + \frac{x \ln x}{x^4}) }{x + \left( \frac{2}{3} \right)^x}...
11. ### Epsilon-delta proof for limits (multivariable)

Homework Statement : the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3). f(x,y) is defined as (2y2+x, -2x+7, x+y). [/B] The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry...
12. ### I Cauchy's Integral Test for Convergence

Hello, I am want to prove that: $$\sum_{1}^{\infty} \frac{1}{n^{2} + 1} < \frac{1}{2} + \frac{1}{4}\pi$$ Cauchy's Convergence Integral If a function decreases as n tends to get large, say f(x), we can obtain decreasing functions of x, such that:  f(\nu - 1) \geqslant f(x) \geqslant...
13. ### Help with Epsilon Delta Proof of Multivariable Limit

Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus cant ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...
14. ### I Help With Epsilon Delta Proof of multivariable limit

I realized I placed this in the wrong forum... I will put it in coursework help
15. ### Confusing log limit

Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...
16. ### Sequence Convergence/Divergence Question

Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...
17. ### I Limit of cosh and sinh

Hi I was wondering how you get this when taking the limit of T going to 0 From this expression of S: Please help I don't see how ln infinity goes to uB/KbT (used u to represent the greek letter. And how does the other expression of sinh and cosh approach 1?
18. ### Limit of arccosh x - ln x as x -> infinity

Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...
19. ### Limits of sequences involving factorial statements!

Homework Statement I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit. an = (-2)n / (n!) In solving this problem, I am not allowed to use any form or variation of the Ratio Test. 2. The attempt at a solution I was...
20. ### Help with Multi-variable Limits

Homework Statement Evaluate or show that the limit does not exist: \lim_{(x,y) \to (0,0)}\frac{ 2x^{4} + 5y^{3} }{8x^{2}-9y^{3}} \lim_{(x,y) \to (0,-2)}\frac{ xy+2x }{3x^{2}+(y+2)^{2}} Homework Equations The Attempt at a Solution So the first one is indeterminate and cannot be...
21. ### Differentiability of a function -- question on bounding

Homework Statement I need to see if the function defined as ##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...
22. ### Limits problem

hi I don't understand how to do one type of homework problem, here's an example of the type: If limit of f(X)/X = 1 as X ->0 evaluate the limit f(X) as X->0
23. ### Finding limits to a piecewise function (3 pieces)

Homework Statement f(x)=-2 when x<1 =3 when x=1 =x-3 when x>1 find the limit at 1 from the left and right sides and at 1. Homework Equations The Attempt at a Solution limit for x when approaching 1 from the left is -2 limit for x when approaching 1 from the right is -2 -I'm not...
24. ### How do I prove that both are equivalent limits

Homework Statement If k is a positive integer, then show that ##\lim_{x\to\infty} (1+\frac{k}{x})^x = \lim_{x\to 0} (1+kx)^\frac{1}{x}## Homework Equations L'Hopitals rule, Taylor's expansion The Attempt at a Solution How should I begin? Should I prove that both has the same limit, or is...
25. ### Unusual Limit involving e

This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{-x}dx=1. Well, when we do Integration by parts we will have let u = x du = dx dv = e^{-x}dx v =...
26. ### Implications of varying the definition of the derivative?

I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...
27. ### Limit of a sequence problem

Homework Statement Consider the sequence given by b_{n} = n - \sqrt{n^{2} + 2n}. Taking (1/n) \rightarrow 0 as given, and using both the Algebraic Limit Theorem and the result in Exercise 2.3.1 (That if (x_n) \rightarrow 0 show that (\sqrt{x_n}) \rightarrow 0), show \lim b_{n} exists and find...
28. ### Prove that function tends to 0 everywhere in this interval

Homework Statement From Spivak: Suppose that ##A_{n}## is, for each natural number ##n##, some finite set of numbers in ##[0,1]##, and that ##A_n## and ##A_m## have no members in common if ##m\neq n##. Define f as follows: ##f(x) = \frac{1}{n}##, if ##x \in A_n## ##f(x) = 0##, if ##x \notin...
29. ### Finding the limit of lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1

Hi, I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit? lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo w and wo are both two points within the same plane.
30. ### Evaluate ##\lim_{x\to 0} \frac{2^x-1-x\log_e2}{x^2}##

Homework Statement Evaluate ##\lim_{x\to 0} \frac{2^x-1-x\log_e2}{x^2}## without using L'Hospital's rule or expansion of the series. Answer is given to be = ##\frac{(\log_e (2))^2}{2}## Homework Equations Squeeze play theorem/ Sandwich theorem, some algebraic manipulations and standard...