I am beginning my study of calculus and I'm not sure I understand limits. What is the difference between to limit of a function as it approaches infinity and the maximum of its graph? When finding the limit of a graph as it approaches a certain number, it seems that most of the time you can simply put the number in the function and solve. What are the cases when this is not true? What is the general procedure for finding the limit of a function?

Originally posted by LionHearted
I am beginning my study of calculus and I'm not sure I understand limits. What is the difference between to limit of a function as it approaches infinity and the maximum of its graph? When finding the limit of a graph as it approaches a certain number, it seems that most of the time you can simply put the number in the function and solve. What are the cases when this is not true? What is the general procedure for finding the limit of a function?
For smooth, continuous functions, you certainly can just plug in the argument to get the limit of the function at that argument.

However, some functions are not so well-behaved. Consider, for example:

f(x) = (x^2 - 9) / (x - 3)

If you graph this function, you'll clearly see that the function approaches the value 6 at x = 3. However, the function does not have a defined value at x = 3. You can't just plug in x = 3.

There is no general "rule" for finding limits -- there are a variety of methods that work in various circumstances. For this simple problem, one would use L'Hopital's Rule, which is expressed here:

http://www.npac.syr.edu/REU/reu94/williams/ch3/subsection3_1_6.html

Essentially, you can take the derivatives of the numerator and denominator of a fraction and form a new fraction that has the same limit. For the function f(x) I gave above, you can do the derivatives of the top and bottom easily: d/dx( x^2 - 9 ) = 2x and d/dx( x - 3 ) = 1. Then you can form a new fraction out of those derivatives; it would be 2x / 1. The limit of this new function as x approaches 3 is now simple -- it's 2(3) / 1, or 6.

- Warren

As always, Warren provided an excellent answer. It is a sad fact that limits can be quite difficult to find.
Originally posted by LionHearted
What is the difference between to limit of a function as it approaches infinity and the maximum of its graph?
I'll assume that you meant to write "between the limit" and not "between to limit". In general, there is no connection between the maximum value of a function and the limit of the function as the independant variable approaches infinity. Here are some examples to think about:

The maximum value of f(x) = 1/(x^2+1) is 1 (at x =0) and the limit as x-> inifinity is 0.

The function f(x) = x^2/(x^2+1) doesn't have a maximum value (but it does have a minimum value), and the limit as x-> infinity is 1.

The function f(x) = sin(x) has as maximum value of 1 (at x = (2n+1/2)*Pi) but the limit as x-> infinity is undefined.

the maxima of the function is the limit at infinity if the maxima occurs at infinity. conside f(x)=1/x has a minima at infinity and maxima at 0 on the interval (0,infinity)

now consider f(x)= (x-3)/(x-3)
the function will look similar to y=1, except at x=3 the denominator becomes 0 and the function becomes undedfined at x=3, and continuous everwhere except at x=3, it is said to have a point discontinuity.

so to answer the question about the limit at infinity and maxima, yes they can be the same thing if and only if the maxima occurs at infinity

eg. y=x