Textbook Notes (280,000)

CA (170,000)

UTSC (20,000)

Mathematics (50)

MATA32H3 (3)

Raymond Grinnell (2)

Chapter 10

# MATA32H3 Chapter Notes - Chapter 10: Bes, Netherlands Institute For Art History, Great News

by OC650170

Department

MathematicsCourse Code

MATA32H3Professor

Raymond GrinnellChapter

10This

**preview**shows pages 1-2. to view the full**6 pages of the document.**LIMITS (week 3 notes)

Topic One: Visual concept: The idea of a limit is us trying to figure out what the graph looks like, what point it hits as it

approaches the left and the right. If the graph approaches the same y-value from the left as it does from the right, then the

limit exists. However, if the graph approaches a different y-value from the left compared to the right, then the limit does

not exist (we write DNE for short).

Ex1:

In this example above, we could be asked to find:

lim

x→ 2

f(x)

; this math language means, “please find the limit (or y-

value) of this function f(x) as x approaches 2”. You must check the graph from the left and the right to see if it approaches

the same value in order for the limit to exist. If I look from the left – that is, as x moves from the left (lets say negative 1

ish) towards the point of interest of x being 2, we see it’s hitting a y-value of 7. If I look from the right – that is, as x moves

from the right (let’s say looking from x is 3) towards the point of interest of x being 2, we see again we’re hitting a y-value

of 7. Since we approach the y-value 7 from both the left and the right then we can securely say;

lim

x→ 2

f(x)=7

Ex2:

Here too, the limit exists despite that solid blue dot. Because we have a the graph

clearly approaching a y – value (let’s guess that it’s about y = 3 since this graph is kind of crappy) from the left and right, it

doesn’t matter that it’s an open dot (remember, open dot means we can’t touch that specific value of y = 3), we are still

approaching that y –value of 3. Therefore I can clearly state that the:

lim

x→ 2

f(x)=3

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

Note 1: The limit is different from finding

f

(

2

)

.

Being asked to find

f

(

2

)

means that whether it’s a closed or open

dot MATTERS. It must be a closed dot for it to be the true answer. The way to find

f

(

2

)

is to recognize that the 2

represents the x-value and we want the y-value when x = 2. Because we have an open dot and closed dot at x = 2, in this

case we take the closed dot, which is a y-value of 4 let’s say (crappy graph..). So this time we say:

f

(

2

)

=4

Note 2: Based on the previous note, if we go back to the first graph where we said:

lim

x→ 2

f(x)=7

, we could also be

asked for

f

(

2

)

.

Since there is no closed dot or solid line through x is 2, then we say

f

(

2

)

=DNE

(does not exist).

Ex3:

In this example, we clearly have a limit. We can say that :

lim

x→ 0

f(x)=1

since as x approaches zero (again

approaching and touching x are different ideas) from the left and from the

right, we approach the y-value of 1. We also can claim

f

(

0

)

=1

, since the

graph at that exact point actually has a value of y = 1.

Ex4:

In this example, we might be asked for

lim

x→ 2

f(x).

In this case, the limit does not exist since from the left we

approach a y-value of 6, but from the right we approach a y-value of 8. However, if we are asked for

f

(

2

)

,

we can say

f

(

2

)

=6

since we have a closed dot, or exact y-value, at the moment x is 2.

###### You're Reading a Preview

Unlock to view full version