Calculus Chapter One Sec 1.2 Limits and Their Properties.

Calculus Chapter One

Sec 1.2Limits and Their Properties

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.


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.


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


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.


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?


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.


We are taking lim b(x), where b(x) X→3

is the breeze that you feel when your nose is at point x.


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 2.9 2.99 2.9999 2.9999 2.99999

Breeze 6 6.7 6.92 6.991 6.999993


It looks like the breeze is approaching 7 mph as your nose approaches the fan.

So, we say that the lim b(x)=7x→3

“The limit of b of x as x approaches 3 is 7.”


Example 1: lim x^2-7x^3+5x→1

Substitute x=1Answer = -1


Example 2: lim 1 x→2 (x-2)^2

x≠2 so try values closer and closer to 2

if x=0→1/4if x=1→1if x=1.5→4if x=1.75→16if x=1.95→400if x=1.9567→533.36


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.


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.


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.


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!!


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.


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.


General Procedures for Taking a Limit

We want to take lim f(x) for some x→b

function f(x).


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.


“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.


Examples 5-6Examples 7-8No 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


Classwork (page 53)#4, 6, 8, 10, 12, 14, 16, 18, 20

Homework (page 53)#3, 5, 7, 9, 11, 13, 15, 17, 19

Calculus Chapter One Sec 1.2 Limits and Their Properties.

Documents

Transcript of Calculus Chapter One Sec 1.2 Limits and Their Properties.