Chapter 2 Limits and Continuity 2.1 Limits (an intuitive approach) Many ideas of calculus originated with the following two geometric problems:

Traditionally, that portion of calculus arising from the tangent line problem is called differential calculus and that arising from the area problem is called integral calculus Tangent lines and limits Areas and limits Decimals and limits*

Limits The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example, let us examine the behavior of the function for x values closer and closer to 2. We can see that the values of f(x) get closer and closer to 3 as values of x are selected closer and closer to 2 on either side of 2.

We describe this by saying that the “limit of is 3 as x approaches 2 from either side”, and we write Limits

Note: Since x is different from a, the value of f at a or even whether f is defined at a, has not bearing on the limit L Limits (An Informal View)

Ex: Use numerical evidence to make a conjecture about the value of

Solution: Although the function is undefined at x=1, this has no bearing on the limit. If we take the x-values approaching 1 from both sides (0.99, 0.999, , , , , 1.001, 1.01), then the corresponding f(x) value appear to get closer and closer to 2. Hence we conjecture that This is consistent with the graph of f. We will learn how to obtain the result algebraically in next section.

One-sided Limits For example: consider the function

As x approaches 0 from the right, f(x) approaches 1, and similarly as x approaches 0 from the left, f(x) approaches -1. We denote this by Here “+” indicates a limit from the right and “-” indicates a limit from the left.

One-Sided Limits

The relation between one-sided limits and two-sided limits In general, there is no guarantee that a function f will have a two-sided limit at a given point. In this case, we say that does not exist. Similarly for one-sided limits. Here we state the relation without formal proof

Ex: for the functions in the slide, find the one-sided and two sided limits at x=a if they exists.

Solution: The functions in all three figures have the same one-sided limits as x->a, since the function are identical, except at x=a. These limits are In all three cases the two-sided limit does not exist at x->a since the one-sided limits are not equal.

Infinite Limits Sometime one-side or two-sided limits fail to exist because the values of the function increase or decrease without bound. For example, consider We describe this behaviors by writing

Infinite Limits (An Informal View)

Vertical Asymptotes If any the following situations occur: Then the line x=a is called a vertical asymptote of the curve y=f(x)

Ex: For the function graphed in the next slide, find all the one-sided limits and two-sided limits at x=4. Furthermore, find the function value at x=4

Ex: For the function graphed in the next slide, find all the one-sided limits and two-sided limits at x=0. Furthermore, find the function value at x=0