Calculus Chapter One Sec 1.2 Limits and Their Properties

What is a “limit”? What if someone says you are “pushing them to the limit”? Usually (in English) the word limit is used to mean a boundary beyond which one cannot go.

Limits and Their Properties Think about a game warden who catches a hunter. The hunter might say he “caught the limit” or “shot the limit”. The number can approach the limit and even reach it, but it cannot exceed it.

Limits and Their Properties In math, a limit is the number that a function approaches as the values of x plugged into the function approach a fixed number.

Limits and Their Properties For example, suppose you move your nose toward a fan. Imagine your nose at position x and the fan at position 3, 3 feet from the origin. Draw example.

Limits and Their Properties We want to know what happens as x get really close to 3. What happens are your nose approaches 3, getting closer and closer, without ever really reaching 3?

Limits and Their Properties You feel the breeze stronger as x gets closer to 3. We want to know what happens to the amount of the breeze as you approach 3.

Limits and Their Properties We are taking lim b(x), where b(x) X→3 is the breeze that you feel when your nose is at point x.

Limits and Their Properties Say you feel a breeze of 6 mph when x=2.9 and the breeze increases as you move your nose toward the fan as in the chart: Nose Position Breeze

Limits and Their Properties It looks like the breeze is approaching 7 mph as your nose approaches the fan. So, we say that the lim b(x)=7 x→3 “The limit of b of x as x approaches 3 is 7.”

Limits and Their Properties Example 1: lim x^2-7x^3+5 x→1x→1 Substitute x=1 Answer = -1

Limits and Their Properties Example 2: lim 1 x→2 (x-2)^2  x≠2 so try values closer and closer to 2 if x=0→1/4 if x=1→1 if x=1.5→4 if x=1.75→16 if x=1.95→400 if x=1.9567→533.36

Limits and Their Properties As x approaches 2, (x-2)^2 gets very small. But 1 divided by a TINY number is a very LARGE number. So, as x approaches 2, 1/(x-2)^2 approaches infinity (∞). Since ∞ is not a real number, we say that the limit does not exist.

Limits and Their Properties Example 2 shows that we can have limits that are so big (bigger than any real number) that the limit does not exist. We can have limits that don’t exist for other reasons.

Limits and Their Properties Example 3 lim sin(1/x) x→0  As x gets smaller and smaller, 1/x gets bigger and bigger. But the sin (1/x) will always be between 1 and -1, since the sine of any number is between 1 and -1.

Limits and Their Properties  As 1/x gets bigger and bigger, sin (1/x) oscillates between 1 and -1 faster and faster.  It goes crazy and is not getting closer and closer to any one number.  It does not zero in on any one value.  Therefore, it has NO LIMIT!!

Limits and Their Properties Think of it as someone who is “falling in love” and can’t commit. They want to play the field….a perpetual swinger!!! See graph on page 50.

Limits and Their Properties Example 4 (Hard limits can be easy at other points.) lim sin (1/x) x→2/π  Answer: 1  The limit of sin (1/x) has problems only as x→0, not when x approaches other values.

Limits and Their Properties General Procedures for Taking a Limit We want to take lim f(x) for some x→b function f(x).

Limits and Their Properties Always plug b into the function. If you get a number (that doesn’t have 0 in the denominator or a negative number inside a square root) and if the function is not one of those weird functions that changes its definition at the point B, then the number f(b) is the limit.

Limits and Their Properties “Plugging in” always works for a polynomial. It usually works for almost any function as long as the point you are plugging in doesn’t have you divide by zero. This is called DIRECT SUBSTITUTION.

Limits and Their Properties Examples 5-6 Examples 7-8 No problem with 0 in the numerator. Worry if it is in the denominator. If 0 is in the denominator…try to simplify the fraction, hoping to get rid of one of the zeros. Examples 9 and 10

Limits and Their Properties Classwork (page 53) #4, 6, 8, 10, 12, 14, 16, 18, 20 Homework (page 53) #3, 5, 7, 9, 11, 13, 15, 17, 19