Math 137 Week 3 Notes – Dr. Paula Smith
I.
The limit of a function.
A.
Motivation
1.
(High school) algebra studied numbers operated on by functions; (university) calculus studies
functions operated on by the limit, differential operator, and integral.
2.
In this course, we will study Differential Calculus, concerning derivatives; a
derivative
at
point
α
can be thought of as the slope of a curve at that point.
3.
Finding the slope of a straight line f(x) is easy – take any two distinct points of the line
(
α
,f(
α
)) and (x,f(x)); slope is (f(x) – f(
α
)) / (x –
α
).
4.
Slope of a curved line f(x) is confusing – which two points to take?
5.
Big idea: the slope of a curve at point (
α
,f(
α
)) is the slope of the straight line tangent to f at
that point.
6.
We use
limits
to determine that slope.
B.
Intuitive ideas
1.
As x
→
a, f(x)
→
L – “as x approaches a (from either side), f(x) approaches the limit L.”
2.
We can make f(x) as close to L as we like by taking x to be su
ﬃ
ciently close to a (though not
equal to a).
3.
We can make f(x)
−
L as small as we like by taking x so that x
−
a is su
ﬃ
ciently small but
not equal to 0.
4.
Notation:
lim
x
→
a
f(x) = L
a.
Be careful; lim
x
→
a
f(x) and f(a) are not always the same.
One might exist while the
other doesn’t.
C.
Onesided limits
1.
It can be easier to think of limits restricted to just one side of
a
.
2.
Notation:
a)
If x > a, lim
x
→
a+
f(x) = L.
b)
If x < a, lim
x
→
a
f(x) = L.
c)
Note that the sign indicating the side is behind
a
; so we might have lim
x
→
4
f(x) = L
or lim
x
→
4+
f(x) = L if a = – 4.
3.
For lim
x
→
a
f(x) to exist, lim
x
→
a+
f(x) = L must exist, lim
x
→
a
f(x) = M must exist, and L must
equal M.
Then lim
x
→
a
f(x) = L.
D.
What can go wrong?
1.
jump discontinuity:
on the left side of
α
f may approach the value M, and on the right side of
α
, f may approach the value N, but M
≠
N.
Since f doesn’t tend to the same value, the limit of f
DNE.
2.
Infinite wiggle:
f(x) = sin(
π
/x) at x approaches 0
3.
Infinity:
f(x) = 1/(x – 1) as x approaches 1
E.
Hence for the limit of f to exist at point
α
, the
lefthand limit
lim
x
→α

f(x) must exist (equal a
real number L), the
righthand
limit
lim
x
→α
+
f(x) must exist, and both must equal the same value
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F.
Mathematically precise de
fi
nition
Lim
x
→
a
f(x) = L if for every
ε
> 0 there is a corresponding
δ
> 0 such that 0 < x
−
a <
δ
implies
f(x)
−
L <
ε
.
G.
Examples
1. Use the precise de
fi
nition of limit to prove that lim
x
→
2
(3x
−
7) =
−
1.
Strategy
:
First we substitute a = 2, f(x) = 3x
−
7, L =
−
1 in the de
fi
nition. We need to
fi
nd a
formula for
δ
in terms of
ε
such that 0 < x
−
2 <
δ
implies (3x
−
7)
−
(
−
1) <
ε
.
Notice that (3x
−
7)
−
(
−
1) = 3x – 6 = 3x – 2 .
If x
−
2 <
δ
, then 3x – 2 < 3
δ
.
This will be less than
ε
if
3
δ
(
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 Spring '08
 SPEZIALE
 Math, Calculus, Algebra, Limit, Limit of a function, Dr. Paula Smith

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