School

Simon Fraser University
Department

Mathematics

Course Code

MATH 157

Professor

Shiva Gol Tabaghi

Review Notes

1. Limit of a function: we write

ax

Lxf

)(lim

and say “the limit of f(x), as x approaches a, equals L” , if we can make the value of f(x)

arbitrary close to L (as close to L as we like) by taking x to be sufficiently close to a (on

either side of a) but not equal to a.

Fact:

axaxax

LxfandLxfLxf ))(lim)(lim()(lim

Special Limits

0

1

sin

lim71828.2.)

1

1(lim

xn x

x

ee

n

n

2. Limit definition of Continuity of a function. A function is continuous at a number a if

ax

afxf

)()(lim

.

Note. a belongs to the domain of f . b)

ax

xf

)(lim

exist . c)

ax

afxf

)()(lim

Discontinuity of a function at a. If

1) f is defined on an open interval containing a, except perhaps at a, and

2) f is not continuous at a

we say that f is discontinuous at a.

Theorem: Intermediate value theorem. Suppose that f is continuous on the closed interval

[a,b] and let N be any number between f(a) and f(b), where f(a)

f(b). Then there exists a

number c in (a,b) such that f(c)=N.

Theorem: Existence of Zeros of a continuous function. If f is a continuous function on a

closed interval [a,b], and if f(a) and f(b) have opposite signs, then there is at least on solution

of the equation f(x)=0 in the interval (a,b).

f

(

a

)

a

b

f

(

b

)

3. Limit Definition of Derivative at the point (a, f(a)). The derivative of a function f at a

defined as

0

)()(

lim

hh

afhaf

.

It is also denoted by

)(xf

and also called instantaneous rate of change. The derivative of a

function at a point x=a is the slope of the tangent line to the graph of f at (a,f(a)).

4. Differentiation Rules:

(a) General formulas:

0)( c

dx

d

)(.))(( xf

dx

d

cxcf

dx

d

1

)(

nn nxx

dx

d

)()())()(( xg

dx

d

xf

dx

d

xgxf

dx

d

)()()()()())()(( ruleproductxfxgxgxfxgxf

dx

d

2

)]([

]

)(

)(

[xg

gfgf

xg

xf

dx

d

)()()).(())](([ rulechainxgxgfxgf

dx

d

x

x

x

dx

d

(b) Exponential and Logarithmic Functions:

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