What is Limit definition: Definition and 57 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. D

    Solving for derivative of e^x using limit definition

    Homework Statement I want to solve for the derivative of e^x using the limit definition. Homework Equations http://www.math.hmc.edu/calculus/tutorials/limit_definition/img10.png The Attempt at a Solution obviously the derivative of e^x is itself, so i konw the answer. i just...
  2. H

    Using the limit definition (f(x+h)-f(x)\h

    I have to differentiate some problems using the limit definition (f(x+h)-f(x)\h), but Iam having some trouble on a couple. 1. Square root of 30 is 0 because its a constant correct? 2. R(t) = 5t^(-3/5) <---- I have tried this problem many times with the definition, but I can't come up with...
  3. K

    Prove by Beta and Sigma limit definition

    Please any help. Use the symbols \beta and \sigma definition of limit to prove that limit (x,y)\Longrightarrow(0,0)x+y/x\x^2+y\y^2=0
  4. Z

    Can X_n Converge to X in Probability if it Doesn't Meet Definitions 2 or 3?

    First 3 definitions: \begin{array}{l} \left( 1 \right)\mathop {\lim }\limits_{n \to \infty } P\left( {\left| {X_n - X} \right| > \varepsilon } \right) = 0 \\ \left( 2 \right)P\left( {\mathop {\lim }\limits_{n \to \infty } X_n = X} \right) = 1 \\ \left( 3 \right)\mathop {\lim...
  5. M

    Can someone explain mimits and how it relates to the formal limit definition?

    i didnt understand delta definiton anyone can explain?
  6. F

    Is This a Correct Definition of a Limit in Calculus?

    show by example that the stament for all d>0 there exists e>0 such that o<|x-a|<d -> |f(x）-L|<e is Not the definition of limit. would somebody give me a hint?
  7. F

    Prove the quotient theorem using the limit definition?

    Hello guys! I'm new here! Well, it feels like this forum is cool and interesting! Can anyone help me here? =) How do you prove the quotient theorem using the limit definition? (Given a limit of f of x as x approaches a is A and a limit of g of x as x approaces a is B).
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