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.
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...
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...
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?
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).