1.4. Example: Substituting numbers can suggest the wrong answer. The previous example
shows that our rst denition of limit is not very precise, because it says x close to a, but how close is
close enough? Suppose we had taken the function
g(x) =
101 000x

109
6.3.3. Calculation of the length of curves
Consider an arbitrary curve in the plane,
described by a function y = f (x)
f (x)
What is the length L of this curve
between, say, x = a and x = b ?
split the x-axis into small steps of size h
e.g. xi = a +

54
4.3. Denitions of hyperbolic functions
4.3.1. Denition of hyperbolic sine and hyperbolic cosine
Option I. denition via dierential equations:
We can dene the hyperbolic sine sinh(z) and hyperbolic cosine cosh(z) as the solutions of the
following equatio

Consider the limit
limx
af(x)g(x)
If both the numerator and the denominator are finite at a and g(a)
limx
=0, then
af(x)g(x)=f(a)g(a)
Example
limx
3x+2x2+1=510=2.
But what happens if both the numerator and the denominator tend to 0? It is not clear
what t

Antiderivatives
Let f(x) be continuous on [a b]. If G(x) is continuous on [a
b), thenG is called an antiderivative of f.
b] and G (x)=f(x) for all x (a
We can construct antiderivatives by integrating. The function
F(x)=
axf(t)dt
is an antiderivative for f

13.2. Theorem (Chain Rule). If f and g are dierentiable, so is the composition f g.
The derivative of f g is given by
(f g) (x) = f (g(x) g (x).
The chain rule tells you how to nd the derivative of the composition f g of two functions f and g
provided you

So this is our recipe for constructing the tangent through P : pick another point Q on the graph, nd the
line through P and Q, and see what happens to this line as you take Q closer and closer to P . The resulting
secants will then get closer and closer t

9.3. Example: a Backward Cosine Sandwich. The Sandwich Theorem says that if the function
g(x) is sandwiched between two functions f (x) and h(x) and the limits of the outside functions f and h exist
and are equal, then the limit of the inside function g e

If you dont remember the geometric sum formula, then you could also just verify (19) by carefully multiplying
both sides with x a. For instance, when n = 3 you would get
x (x2 + xa + a2 ) = x3
a (x2 + xa + a2 ) =
(x a) (x2 + ax + a2 )
+ax2
ax2
+a2 x
a2 x

Now divide by x a and let x a:
lim
xa
f (x) f (a)
v(x) v(a) u(x) u(a)
= lim u(x)
+
v(a)
xa
xa
xa
xa
(use the limit properties)
u(x) u(a)
v(x) v(a)
+ lim
v(a)
xa
xa
xa
= u(a)v (a) + u (a)v(a),
=
lim u(x)
lim
xa
xa
as claimed. In this last step we have used

range of f
y = f (x)
(x, f (x)
x
domain of f
Figure 3. The graph of a function f . The domain of f consists of all x values at which the function is
dened, and the range consists of all possible values f can have.
m
P1
y1
1
y1 y0
y0
P0
x1 x0
n
x0
x1
Figur

CHAPTER 1
Numbers and Functions
The subject of this course is functions of one real variable so we begin by wondering what a real number
really is, and then, in the next section, what a function is.
1. What is a number?
1.1. Dierent kinds of numbers. The

f (x) < 0
f (x) > 0
f (x) < 0
1
1
1
x
1
f (x) < 0
f (x) < 0 for all x
but f is not decreasing
since f (1) > f (1)!
6.2. Example: the hyperbola y = 1/x. The derivative of the function f (x) = 1/x = x1 is
1
x2
which is always negative. You would therefore t

Remember the trick and divide top and bottom by x2 , and you get
3 + 3/x2
3x2 + 3
= lim
x 5 + 7/x 39/x2
x 5x2 + 7x 39
limx 3 + 3/x2
=
limx 5 + 7/x 39/x2
3
=
5
Here we have used the limit properties (P ) to break the limit down into little pieces like limx