What is Limit at infinity: Definition and 38 Discussions

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

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  1. A

    Evaluate the limit of this harmonic series as n tends to infinity

    To use the formula above, I have to prove that $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow \infty}\left(\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+........\frac{1}{n}}{n^2}\right)=1$$ To prove so, I tried using L'Hopital's Rule: $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow...
  2. 0

    I A question about limits and infinity

    I hope I can make this question clear enough. When we have a function such as f(x) = 1/x and calculate the side limits at x = 0, the right side goes to positive infinity. The left side goes to negative infinity. In calculus we are pluggin in values closer and closer to zero and seeing what the...
  3. R

    I How can we define a limit approaching negative infinity?

    I have the following definition: $$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$
  4. yucheng

    Computing a limit involving a square root: what is wrong?

    My attempt: \begin{align} \lim\limits_{n \to \infty} \sqrt{n^2 + n} - n &= n\sqrt{1+\frac{1}{n}} -n\\ &=n - n\\ &= 0\\ \end{align} I think the issue is at (1)-(2) For comparison, here is Rudin's solution
  5. K

    I Determine the limit in a Markov process over time

    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...
  6. L

    I Simplifying limit with Stirling approximation

    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...
  7. Physics lover

    Simplified Limit Calculation for (1-e^(1-x/(1+x))x)/(1/x)

    I simplified somewhat and got (1/e-(1-x/(1+x))x)/(1/x) So i can't find that it is 0/0 form so tried by applyying L'Hospitale,But it just became complicated.So need help.
  8. S

    I Precise intuition about limits and infinitesimals

    I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals. For example, I know...
  9. EastWindBreaks

    The Halpin Tsai relation to Voigt model

    Homework Statement Homework Equations Halpin Tsai equation:[/B] Voigt model: P*=VfPf+VmPm Vf= volume of fiber/ volume of composite Vm= volume of matrix / volume of composite ζ= estimated parameter Pf, Pm= fiber and matrix properties V_v=1-Vf-Vm The Attempt at a Solution Since there is no...
  10. E

    Then, as ##b## goes to 0, can you find the limit of each factor separately?

    Homework Statement a. Compute the limit for f(x) as b goes to 0 Homework Equations $$f(x) = \frac{(a+bx)^{1-1/b}}{b-1}$$ ##a \in R##, ##b\in R##, ##x\in R## The Attempt at a Solution ##a+bx## goes to ##a## ##1/b## goes to ##\infty## so ##1-1/b## goes to ##-\infty## ##(a+bx)^{1-1/b}## then goes...
  11. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Homework Statement ##a_1## = 3 ##a_{n+1}## = ##\frac{2}{3} a_n + \frac{1}{4} ## Homework Equations [/B]The Attempt at a Solution Sequence i got : ## a_1, a_2, a_3, a_4, a_5 ## ## 3, \frac{9}{4}, \frac{7}{4}, \frac{17}{12}, \frac{43}{36} ## I tried to find the formula of ##a_n## ##a_n =...
  12. R

    Limit of x^α.sin²(x)/(x+1) as x approaches infinity is?

    Homework Statement : Find [/B] limx->∞(xα(sin2x!)/(x+1) α∈(0,1) Options are: a)0 b)1 c)inifinity d)does not existHomework Equations : -[/B]The Attempt at a Solution : limx->∞(xαsin2x!)/(x+1)[/B] Dividing the numerator and denominator by xα, we have:limx->∞sin2x!)/(x1-α+x-α) clearly x−αis tending...
  13. A

    Finding the limit at infinity

    Homework Statement Finding the value of the limit: $$\lim_{t\to +\infty} t+\frac{1-\sqrt{1+a^2t^2}}{a}$$ ##a## is just a costant The Attempt at a Solution At first sight I had thought that the limit was ##\infty## but then I realized that there is an indeterminate form ##\infty - \infty##. I...
  14. S

    Evaluating limit at infinity by Maclaurin series

    Homework Statement I've begun going through Boas' Math Methods in the Physical Sciences and am stuck on problem 1.15.25. The problem is to evaluate ## \lim_{x\to \infty } x^n e^{-x} ## By using the Maclaurin expansion for ##e^{x}##. Homework Equations We know the Maclaurin expansion for the...
  15. binbagsss

    Elliptic function - does limit at infinity exist?

    Homework Statement I am wanting to show that ##lim_{z\to\infty} f(z)=c## does not exist for ##c \in C##, ##C## the complex plane, where ##f## is non-constant periodic meromorphic function. (elliptic) Homework EquationsThe Attempt at a Solution So I want to proove this is not true ...
  16. A

    Limit at infinity with radicals

    Homework Statement lim as x tends to -∞ (x)^3/5 - (x)^1/5 Homework EquationsThe 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 x^3...
  17. G

    MHB Show $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$

    How do you show that $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$ What I tried: $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1} =\lim_{x \to \infty} \frac{50+100/x^{11}}{1+1/x^{5}+1/x^{11}} = \frac{50+0}{1+0+0} = 50.$ But this is wrong. (Angry)
  18. P

    Solve Limit at Infinity: $(-1)^n \sqrt{n+1}/n$

    Homework Statement $$\lim_{x\to\infty} \dfrac{(-1)^n\sqrt{n+1}}{n}$$ Homework Equations 3. The Attempt at a Solution [/B] This is what I managed to do but I just wanted to verify that this is the correct way of solving it, I'm mainly concerned about the fact that I took the absolute value...
  19. terryds

    What is the Limit at Infinity for (2^x-5^x) / (3^x+5^x)?

    Homework Statement lim x->∞ (2^x-5^x) / (3^x+5^x) Choices : a. -1 b. -2/3 c. 1 d. 6 e. 25 2. The attempt at a solution Hmmm.. I really have no idea about this.. This is an unusual problem.. Please tell me...
  20. J

    MHB Left hand and right hand limit at infinity

    Show that $\lim_{{x}\to{0}}$ $\frac{1}{x}$ does not exist? Please tell me how to make LHL and RHL for this? Explain me all steps used?
  21. A

    MHB Solve Limit at Infinity Problem: Tips & Help

    Hi, I've been doing limit problems, and just got to this problem and I can't solve it. I would love some tips; you don't have to solve my problem. Screenshot by Lightshot
  22. F

    MHB When taking the limit at infinity, is this allowed?

    In this video of Laplace transforms the equation \lim_{t \to \infty}\frac{te^{-st}}{-s} is said to be 0. I'm not sure I agree with the reasoning. It says it's because e^t grows faster than t; can you treat infinity like that? For example could you say \lim_{x \to \infty}\frac{x}{x^2}=0? I...
  23. P

    What is the limit at infinity for the given expression?

    1. Evaluate lim x\rightarrow-\infty \sqrt{x^2+x+1}+x.The answer is -\frac{1}{2}. Homework Equations None. The Attempt at a Solution I multiplied by the conjugate first, so it turns into lim x\rightarrow-\infty \frac{(x^2+x+1)-x^2}{\sqrt{x^2+x+1}-x} = lim x\rightarrow-\infty...
  24. Petrus

    MHB Horizontal Asymptote of Inverse Tangent Function

    Hello MHB, I got one question, I am currently working with an old exam and I am suposed to draw it with vertican/horizontal lines (and those that are oblique). f(x)=\frac{x}{2}+\tan^{-1}(\frac{1}{x}) for the horizontel line \lim_{x->\infty^{\pm}}\frac{x}{2}+\tan^{-1}(\frac{x}{2}) Is it enough...
  25. C

    Limit at infinity; l'hopital's rule not working as expected

    Homework Statement Lim(t->(inf)) 1/2((t^2)+1) + (ln|(t^2)+1|)/2 - 1/2 Homework Equations N/A (unless L'Hopital's rule can be counted as an equation for this section) The Attempt at a Solution Background: The problem started with: inf ∫(x^3)/((x^2)+1)^2 dx 0 Using partial fraction...
  26. E

    An example of a continuous function in L1 space with no limit at infinity

    Homework Statement I am trying to come up with a continuous function in L1[0,infinity) that doesn't converge to 0 as the function goes out to infinity. Homework Equations I am trying to show an example of an f in L1[0,infinity) (i.e. ∫abs(f) < infinity) where the limit as the function...
  27. D

    MHB Solving Limit at Infinity Problem: x^(2/3) / (log^2(x))

    Hello, I am working on some limit problems and ran into one in which I am lost on how to proceed with: The problem is: lim x -> infinity x^(2/3) x/(log^2(x)) I have a basic understanding of L'Hopital's Rule and attempted to apply it, but just ended up with a confusing mess. I'm assuming I'm...
  28. M

    Understanding Limits at Infinity and Non-Existence of Derivatives

    Homework Statement Hello everyone, I am just new to this forum and also a beginner at calculus. I have a question from my textbook. It's: Find an example of f(x) that satisfies the following conditions : f(x) is differentiable for all x>0; limx->∞f(x) =2; limx->∞f'(x) does not exist...
  29. J

    Limit at infinity, no vert asymptote, why?

    Homework Statement Find the vertical asymptote(n) and evaluate the limit as x \rightarrow n^-, x\rightarrow n^+, or state Does Not Exist. Homework Equations \frac{\sqrt{4x^2+2x+10}-4}{x-1} The Attempt at a Solution I have taken the limits at \pm\infty=2,-2 and understand those are my...
  30. E

    Something weird about limit at infinity?

    Homework Statement This is a problem I came up with when I was doing something similar in Spivak's Calculus; although a simpler version. Suppose, we have f(x)=x^3 and g(x)=x^2 find \lim_{x\rightarrow \infty} f(x)/g(x) Homework Equations N/A The Attempt at a Solution...
  31. K

    Analysis(sequences) proof: multiplying infinite limit at infinity by 0

    Homework Statement Let \stackrel{lim}{_{n \rightarrow \infty}}a_{n} = \infty Let c \in R Prove that \stackrel{lim}{_{n \rightarrow \infty}} ca_{n}= \infty for c>0 (i) - \infty for c<0 (ii) 0 for c=0 (iii) Homework Equations Definition of divergence to infinity (infinite limit at...
  32. M

    Limit at Infinity: Solve Without Squeeze Theorem

    Homework Statement \lim_{x \to \infty} \sqrt{x}\sin\frac{1}{x} Homework Equations I don't think you can use the squeeze theorem here...The Attempt at a Solution So I am just studying for an exam that I have tomorrow and I am going through problems that weren't assigned on our homework set...
  33. M

    Limit at infinity that may or may not be e

    Homework Statement Compute lim as x goes to infinity of (1+1/x^2)^x Homework Equations I know that lim at infinity (1+1/x)^x=e I do not know if that is still valid with the x^2 there. I don't really think it is, but it's throwing me off. The Attempt at a Solution Beyond the...
  34. I

    Proving Limit at Infinity: n^(1/n) = 1

    Homework Statement How can I prove that: \lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1 Isn't \infty^{0} indeterminate? Thanks! Homework Equations The Attempt at a Solution
  35. A

    Find a Function f(x) with Limit at Infinity but no Derivative Limit

    Hello I need to give an example of a function f(x) such that limf(x), as x goes to infinity, exists but the limf'(x) does not. Any help? Thx
  36. M

    Limit at infinity of an electric quadrupole

    This is for my intro physics 2 class Homework Statement Consider the charges Q at (-a, 0), -2Q at (0, 0) and Q at (a, 0). Such a combination of charges, with zero net charge and with zero net dipole moment, is called an electric quadrupole. a. Find the electric field along the x acis, for...
  37. I

    How to Evaluate a Limit at Infinity with Exponential Functions

    Can someone give me a hint on how to evaluate the following limit? \stackrel{lim}{T\rightarrow\infty} (Texp(c/T) - T) I tried multiplying the numerator and denominator by the conjugate (because that sometimes helps) and got: (T^2exp(2c/T) - T^2) / (Texp(c/T) + T) But I'm not sure what I...
  38. R

    What is the Process to Find the Limit at Infinity of (x+(x^2+12x)^1/2)?

    Find the limit as x-> -infinity for (x+(x^2+12x)^1/2) so first of all..i multiply and divide by the conjugent then i get... -12x/(x-(x^2+12x)^1/2) i divide by x in both the nummerator and denominator to get ... -12/1-(1+12/x)^1/2 so the 12/x goes to 0 and the squroot of 1 is 1 so it...
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