The Limit : In the earlier section we looked at some problems & in both problems we had a function (slope in the tangent problem case & average rate of change in the rate of change problem) and we desired to know how that function was behaving at some point x = a .  At this stage of the game we no longer care where the functions came from & we no longer care if we're going to illustrates them down the road again or not. All that we have to know or worry regarding is that we've got these functions and we desire to know something about them.

To answer the questions in the last section we select values of x that got closer & closer to

x = a and we plugged these in the function.  We also ensured that we looked at values of x that were on both the left & the right of x = a .  one time we did it we looked at our table of function values & saw what the function values were approaching as x got closer & closer to x = a and utilized it to guess the value that we were after.

This procedure is called taking a limit and we have some notation for this. For instance the limit notation is,

In this notation we will consider that we always give the function which we're working with and we also give the value of x (or t) that we are moving in towards.

In this section we will take an intuitive approach to limits & try to obtain a feel for what they are and what they can tell us concerning a function. Along with that goal in mind we are not going to get into how we in fact compute limits yet.

Both of the approaches that we are going to use in this section are designed to help us understand just what limits are.  In general we don't typically use the methods in this section to compute limits and in several cases can be very hard to use to even estimate the value of a limit and/or will give the wrong value on occasion.  We will look at actually computing limits in a couple of sections.