Functions and their limits
1.1
1.1.1
Functions
In this course, we will study functions of one real variable.
Example. f (x) = sin(x)
I
x
2
2
x +1
Example. f (x) = |x|
Year 110
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Vocabulary of functions
Example. f (x) =
WEEK 1
if x < 0,
if x Ø 0
Calculus 1 - Year 110
1.1.2
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Exponential functions and their properties
In this section, we study functions of the form
f (x) = ax for some a > 0.
For example, the graph of y = 2x is shown in Figure 1.
Figure 1. The graph of y = 2x
3 4x
1
Example. Sketch the graph for y =
.
2
Example. You are given the sketch of four exponential functions in the following
graph. Compare the size of a, b, c and d.
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Calculus 1 - Year 110
1.1.3
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What is e - episode I
Definition 1.1.1. ‘e’ is the real number such that the slope of the tangent to y = ex
at (0, 1) equals to 1.
Some of you may have seen the following, perhaps, more ‘popular definition’ of the
number e.
Definition 1.1.2 (Alternative definition). ‘e’ is the limit of y =
to infinity.
In other words,
e = lim
xæŒ
3
1
1+
x
4x
3
.
1
1+
x
4x
when x tends
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Calculus 1 - Year 110
1.1.4
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Logarithms - ‘inverse’ to exponentials
Definition 1.1.3. Let a > 0 and a ”= 1. Define loga (x) = k if x = ak .
Example. Sketch the graph of y = log2 (x).
Theorem 1.1.1 (Properties of logarithm). The following are true.
(1) loga (x) + loga (y) = loga (xy).
(2) loga (x) ≠ loga (y) = loga
3 4
x
.
y
(3) loga (xr ) = r loga (x) for any real number r.
(4) (Base change formula) loga b =
logc b
.
logc a
In other words, the logarithm
captures the exponent that you
need to obtain x from base a.
We will explain the shape of this
graph when we discuss ‘inverse
functions’ in the future.
Calculus 1 - Year 110
Proof. [Proof of Base change formula]
Definition 1.1.4. The natural logarithm ln(x) is the ‘logarithm with base e’. i.e.
ln(x) = loge (x).
Example. Solve the equation ln(x) ≠ ln(4x + 2) = 1.
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Calculus 1 - Year 110
1.2
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Inverse functions
The inverse of a function is a function that can ‘undo’ the effect of the original function.
1.2.1
Definition & Examples
Definition 1.2.1. The inverse function of f is the function g such that
g(f (x)) = f (g(x)) = x.
Notation : we use f ≠1 (x) to denote the inverse function of f .
Example. The inverse function of f (x) = x5 is
1.2.2
Graphs of inverse functions
Here we give the sketch of the graphs of
f (x) = x3 + 1 and f ≠1 (x) =
Ô
3
x ≠ 1.
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Calculus 1 - Year 110
1.2.3
One-to-one functions
Unfortunately, not every function has an inverse.
Example (Bad example). What is the inverse function of f (x) = x2 ?
From this example, the following question arises :
‘Why/when does the inverse of a function exist?’
In mathematics, we are not satisfied by just how to compute things. Indeed we are much
more interested in ‘why’. From the above example, we inspected that the issue is that the
above function is ‘two-to-one’. This inspires the following definition.
Definition 1.2.2. A function f is called one-to-one if x ”= y implies f (x) ”= f (y).
It turns out that the ‘one-to-one’ property of a function is equivalent to the existence
of its inverse.
Theorem 1.2.1. The inverse of a function f exists … the function f is one-to-one.
1.2.4
Graphical interpretations of ‘one-to-one’ functions
Theorem 1.2.2. [Horizontal line test] A function f (x) is one-to one … every horizontal
line intersects the graph of y = f (x) in at most one point.
Example. Is the function f (x) = 2x one-to-one?
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Calculus 1 - Year 110
Theorem 1.2.3 (Monotonicity implies one-to-one).
If f (x) is strictly increasing or decreasing, then f (x) is one-to-one (and hence the
inverse of f (x) exists).
Proof. Any strictly increasing or decreasing function will pass the Horizontal line
test and, hence, by Theorem 1.2.2 is one-to-one.
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Calculus 1 - Year 110
1.2.5
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Inverse trigonometric functions
In this section, we discuss the inverses of ‘trigos’ (sine, cosine and tangent). Firstlt, we
recall the graph of y = sin(x).
However, this function is not one-to-one (in particular, HLT fails). To fix this, we are going
to sacrifice parts of the domain.
Domain
sin≠1 (x)
cos≠1 (x)
tan≠1 (x)
Range
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Calculus 1 - Year 110
Example. Sketch the graph of y = arccos(x).
!
"
Example. Simplify the expression cos tan≠1 (x) .
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Calculus 1 - Year 110
1.3
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Limits - the basics
In this section, we develop the language of ‘limits’ that allows us to describe the asymptotic
behaviours of a function, which lies at the subject matter of Calculus.
1.3.1
Intuitive definition
Definition 1.3.1 (‘Definition’). We write lim f (x) = L if the values of f (x) get (arbixæa
trarily) close to L when x gets sufficiently close to a (from both sides) but not equal
to a.
x2 ≠ 4
.
xæ2 x ≠ 2
Example. Compute the limit lim
x2 ≠ 4
(x + 2)(x ≠ 2)
= lim
= lim (x + 2) = 2 + 2 = 4
xæ2 x ≠ 2
xæ2
xæ2
x≠2
Solution. lim
Example. Compute the limit limfi
xæ 2
1 + cos(2x)
.
cos(x)
!
"
Example. Compute the limit lim x tan cos≠1 (x) .
xæ0
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Calculus 1 - Year 110
1.4
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One-sided limits
Definition 1.4.1 (‘Definitions’). We write
(a) lim+ f (x) = L if the values of f (x) get (arbitrarily) close to L when x approaches
xæa
to a from the right, but not equal to a.
(b) lim f (x) = L if the values of f (x) get (arbitrarily) close to L when x apxæa≠
proaches to a from the left, but not equal to a.
Theorem 1.4.1. If lim+ f (x) = lim≠ f (x) and its value equals to L, then lim f (x)
xæa
exists and is equal to L as well.
Example. Let f (x) =
I
x
2
x +1
xæa
xæa
if x < 0
if x Ø 0
. Compute lim+ f (x) and lim≠ f (x).
xæ0
xæ0
Solution.
lim f (x) = lim+ (x2 + 1)
xæ0+
xæ0
=1
lim f (x) = lim≠ x
xæ0≠
xæ0
=0
Example (Absolute values). Compute the limits lim+
xæ0
x
x
and lim≠
.
|x|
xæ0 |x|
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Calculus 1 - Year 110
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x2 ≠ 4x + 3
Example (Trap). Evaluate the limit lim≠ 
.
xæ3
(x ≠ 3)2
Example (Greatest integer function). Compute the limits lim+ [x] and lim≠ [≠x].
xæ2
xæ2
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