CHAPTER 2: LIMITS AND DEFINITION OF DERIVATIVEGroup 3:

Katie Blackburn, Danielle D’Onofrio, Mike Ghen

Definition of A Derivative

3

3

3103310

310)(310

310)

h

h

h

xhx

h

xhx

xya

62

62

62

6662

6)(6)(

6)

2

222

22

2

x

hxh

hhxh

h

xxhxhxhx

h

xxhxhx

xxyb

22

22

22

]112[]1221[

]1)1[(]1)1)[((

1)1()

2

222

22

2

x

xhh

xhxhh

h

xxxhxhhx

h

xhx

xyc

h

xfhxfhx

)()(lim

Conditions of Continuity at “a”

axlim Must exist

F(a) must = axlim

F(a) must exist

Main Limit Theorems

Direct substitution

Simply replace x (or the variable) with the value that x is approaching.

)70(lim 28

xx

Use Algebra (Conjugate method) Use algebra to get rid of 0 in

denominator.

t

tt 3

9lim9

Limits to Infinity

If the degree of the numerator is smaller than that of the denominator Limit f(x)=0

If the degree of the numerator is the same as the denominator Limit f(x)= ratio of the leading coefficients

If the degree of the numerator is larger than that of the denominator Limit f(x)=

01

3lim

xx

31

3lim

x

xx

1lim

2

x

xx

Special Limits

x

xx

sinlim

0

x

xx

1coslim

0