We say lim f (x) exists if
x→a
lim f (x) = lim+ f (x) = L (L 6= ∞).
x→a−
x→a
We say that f (x) is continuous at x = a if three conditions are met:
(i) f (a) is defined
(ii) lim f (x) exists
x→a
(iii) lim f (x) = f (a)
x→a
Types of Discontinuity:
Most limits can be calculated simply by substitution.
Examples:
(i) lim (2x2 + 7x − 2)
x→−1
x+7
x→3 x − 2
(ii) lim
(iii) lim
x→2
(iv) lim
x→2
√
x−7
√
2x + 1
Not so good news: There are discontinuous functions out there!
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Pathological Examples:
• limx→a f (x) is
a number
0
after substitution.
5x
x→1 x − 1
(i) lim
• limx→a f (x) becomes
0
0
after substitution.
x2 − 9
x→3 x − 3
(ii) lim
• limx→a f (x) and a is an endpoint of the domain of f (x).
(ii) lim
x→2
√
x−2
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Summery: To calculate the lim f (x) we substitute x = a unless one of the
x→a
above happens.
Examples: Find the following limits. Is the function continuous?
x2 − 9
(i) lim
x→3 x − 3
√
x+3−2
(ii) lim
x→1
x−1
(x − 1)2 − 1
x→2
x−2
(iii) lim
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(iv)f (x) =
x2 − 2x − 3
x2 + 5x + 4
Example: (piecewise defined 
function)

1 + x x < 2
Consider the function f (x) = 2
x = 2.

 2
x
x>2
What is limx→2 f (x)? Is f (x) continuous at x = 2?
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We say a function is continuous is it is continuous over its domain.
Derivative
First Principal Definition
Recall:
• Instantaneous rate of change of f (x) at x = a is the slope m of the
tangent line at x = a.
• Instantaneous rate of change of f (x) at x = a could be estimated by
taking the slope of the secant line through (a, f (a)) and (a+h, f (a+h)).
• We get a better estimate if we let h → 0.
• In fact m = limh→0
f (a+h)−f (a)
.
h
The instantaneous rate of change of y = f (x) at P = (a, f (a)) is equal to the
slope of the tangent line to the curve y = f (x) at x = a and is also called
the derivative of y = f (x) at x = a and is written f 0 (a).
f (a + h) − f (a)
h→0
h
f 0 (a) = lim
Example:
Consider f (x) = x2 . Find the derivative and the equations of the tangent
lines at x = −1 and x = 2.
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What is f 0 (x)?
The derivative of a function is itself a function of x and we can find it as such
f (x + h) − f (x)
.
h→0
h
f 0 (x) = lim
Examples:
Find f 0 (x) for each case.
1. f (x) = x
2. f (x) = x2
3. f (x) = x3
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Observation?
4. f (x) =
1
x
5. If f (x) = mx + b
Example:
The position of a moving object is given by the position function function
s(t) = 2t2 + 3t + 2 (where s is in m and t in s). What is the velocity function
of this object?
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Does derivative always exists?No (if a not in the domain or f (x) is not continues at a or has a cusp at a or has a vertical tangent or ..)
f (a + h) − f (a)
exists
h→0
h
f (x) is differentiable at x = a if f 0 (a) = lim
Examples:
1. f (x) =
x+2
x−2
2. f (x) =
at x = 2.
3x + 1, x ≤ −1
2 − x2 , x > −1
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(
x
x≥0
3. f (x) = |x| =
.
−x x < 0
Remark: Here is an alternative notation for the derivative. Suppose we
have f (x) = y
df
y 0 = f 0 (x) =
dx
and
df
|x = a.
f 0 (a) =
dx
Do I have to go through this every time I need to derive a function?
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