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Sophie Chrysostomou
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Mathematics

MATA30H3

Sophie Chrysostomou

Winter

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2.1 THE LIMIT OF A FUNCTION AT A POINT
2.1 THE LIMIT OF A FUNCTION AT A POINT
The Tangent Line Problem
DEFINITION: The tangent line (or simply the tangent) to a curve at a
given point is the straight line that ”just touches” the curve at that point, so
that as it passes through the point of tangency, the tangent line has the same
direction as the curve, and in this sense it is the best line approximation to
the curve at that point.
How do we ﬁnd the equation of the tangent line to the graph of a function,
say f(x), at the point (a,f(a))?
.
l
.
. .
0
rise f(x)−f(a)
The slope of the line PQ is the diﬀerence quotient F(x) = run = x−a .
43 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY
Example: Find the slope of the tangent line to the curve of f(x) = x at
the point P(1,1).
x < 1 F(x) x > 1 F(x)
0.9 1.1
0.99 1.01
0.999 1.001
0.9999 1.0001
0.99999 1.00001
→ 1 − → → 1 + →
DEFINITION: If f is a function, and L is a ﬁnite number, we say:
“the limit of f(x), as x approaches a, equals L”, or “f(x) converges to L
as x approaches a” and we write
“ lim f(x) = L” or “f(x) → L as x → a”
x→a
if “the value of f(x) can be made arbitrarily close to L, for all x suﬃciently
near, but not equal to, a.
44 ▯2013 by Sophie Chrysostomou 2.1 THE LIMIT OF A FUNCTION AT A POINT
sinx
EXAMPLE: Investigate the behaviour of f(x) = x near a = 0. Note
that f is not deﬁned at a = 0. Find lim f(x).
x→0
x < 0 f(x) x > 0 f(x)
−0.1 0.1
−0.01 0.01
−0.001 0.001
→ 0− → → 0+ →
2
HOMEWORK: Investigate the behaviour of f(x) = x near a = 2.
45 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY
√
9x − 36x + 36
Now let us investigate the behaviour of f(x − 2 near a = 2.
Conclusions:
x < 2 f(x) x > 2 f(x)
→ → → →
46 ▯2013 by Sophie Chrysostomou 2.2 ONE SIDED LIMITS
2.2 ONE SIDED LIMITS
Consider the following function:
1
x + 1 if x ≥ 1
H(x) = 2
−1 if x < 1
with graph:
3
2
1
What is the lim H(x) ?
x→1
-1 0 1 2 3
-1 .
47 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY
DEFINITION: x→am+f(x) = L where L is a ﬁnite number
is the right limit of f(x) as x approaches a from the right if the values of
f(x) can be made arbitrarily close to L, for all x suﬃciently close to a, but
greater than a.
HOMEWORK: lim f(x) = L where L is a ﬁnite number
x→a −
is the left limit of f(x) as x approaches a from the left
1.) If lx→a+(x) ▯= lx→a−(x), then we say that lix→a(x) does not exist.
2.) If lim f(x) = lim f(x) = L, and L is ﬁnite, then we say that
x→a+ x→a−
x→am f(x) exists and
x→am f(x) = L.
48 ▯2013 by Sophie Chrysostomou 2.3 LIMIT LAWS
2.3 LIMIT LAWS
1. Constant Law If f is the constant function given by f(x) = c for all
x ∈ domf, then lim f(x) = c.
x→a
2.x→am x = a
3. Sum Law If lim f(x) = L and lim g(x) = M and L and M are ﬁnite
x→a x→a
numbers, then
lim(f(x) + g(x)) =
x→a
4. Product Law If lim f(x) = L and lim g(x) = M and L and M are
x→a x→a
ﬁnite numbers, then
lim(f(x)g(x)) =
x→a
49 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY
5. Quotient Law If lix→a(x) = L and lix→a(x) = M ▯= 0 and L and M
are ﬁnite numbers, then
lim(f(x)/g(x)) =
x→a
√ √
6. Root Law lim nx = n a (Note: for n even we require a > 0.)
x→a
There is also a version of the root law that uses one-sided limits, which
stat√s
lim nx = 0 for n even.
x→0+
7. Basic Substitution Law Suppose that f is a polynomial or a rational
function and a ∈ domf, then x→a f(x) = f(a).
x − 1 x − 1
Examples: lim and lim
x→1 x + 1 x→1 x − 1
50 ▯2013 by Sophie Chrysostomou 2.3 LIMIT LAWS
8. If I is an interval containing a and f(x) ≤ g(x) for all x ∈ I, except
possibly at a, then:
x→a f(x) ≤ lix→a(x)
assuming both of these limits exist.
9. Squeeze Law (or Sandwich Theorem) Let I be an open interval
containing a. Suppose that g(x) ≤ f(x) ≤ h(x) for all x ∈ I − {a}. If
lim g(x) = L = lim h(x), then lim f(x) = L.
x→a x→a x→a
1
EXAMPLE: Consider lim x sin 2 .
x→0 x
θ sinθ
HOMEWORK: Show lim = 1 = lim . .
θ→0 sinθ θ→0 θ
(Hint: Show sinθcosθ < θ < tanθ) .
θ tan θ
sin θ
0 θ
Circle of radius 1
51 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY
2.4 INFINITE LIMITS: VERTICAL ASYMPTOTES
(f(x) → ±∞)
1
Consider lim 2.
x→1 (1 − x)
16
14
12
10
8
6
4
2
-3 -2 -1 -2 1 2 3 4 5
Discussions:
A) What is inﬁnity?
B) Is there a diﬀerence between i) lim f(x→a= ∞ and ii) lim f(x) dox→anot
exist (or DNE as most are used to writing).
DEFINITION: If I is an interval containing a, and f is deﬁned for all
x ∈ I, except possibly at a,
x→a f(x) = ∞
means that the value of f(x) can be made arbitrarily large for all x suﬃ-
ciently close to a, but not equal to a.
HOMEWORK: Write similar deﬁnitions for any two of:
x→a f(x) = −∞, lim+f(x) = ∞, lim+f(x) = −∞,
x→a x→a
lim− f(x) = ∞, lim −(x) = −∞
x→a x→a

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