Chapter 12.1 Part Two-Introduction to Limits

Properties of Limits

1.
lim b=b
^x→a

example
lim5=5
^x→2













2.
lim x=a
^x→a

example
lim x=2
^x→2











3.
lim xⁿ=a^n
x→a

example
lim x²=4
^x→2











4.

^x→c

example

<sup>x→8

Operations with Limits

Let f(x)=L
x→a
and
g(x)=M
x→a

Scalar Multiple
lim[c × f(x)]=c × L
x→a</sup>

Sum or Difference
lim[f(x)± g(x)]=L ± M
^x→a

Product
lim[f(x) × g(x)]=L × M
^x→a

Quotient
lim[f(x) ÷ g(x)]= L ÷ M M&ne0
^x→a

Power
lim[f(x)]ⁿ=L^n
x→a

Direct Substitution

is used when:
lim f(x)=f(a)
^x→a

• doesn't work with breaks, holes or asymptotes
  
• would be "false" on a quiz because it is not always true.
  

example
lim x^2
x→4

4²
16

Continuous

• no breaks, holes or asymptotes
  
• draw without lifting pencil
  
• lim f(x)=lim f(x)=f(a)
  ^{x→a+ x→a−}
• if the limit from the left and the limit from the right are both equal to f(a) then it is continuous

not continuous










Piecewise



lim f(x)
x→1−
2(1)-3
-1

lim f(x)
x→1+
-(1)²
-1

Since the limit coming from the right and coming from the left are equal(-1), the limit is -1

if the limits from both sides were different the limit does not exist.

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