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1. Use the following graph of F to nd each limit.

When necessary, state that the limit does not exist.

a . lim x 3

F ( x)e . lim x 4

F ( x)

b . lim x 2

F ( x) f . lim x 6 F ( x)

c . lim x 2

F ( x)g . lim x

F ( x)

d . lim x 5

F ( x)h . lim x

F ( x)

2. Use the following graph of f to nd each limit.

When necessary, state that the limit does not exist. x 3 f ( x)

a . lim x 1

f ( x)e . lim

x 1 f ( x) x 1 f ( x)f .lim

b . lim

x 3+f ( x)g . lim x

f ( x)

c . lim

d . lim x 0

f ( x)h . lim x

f ( x)

3. Eval ate the following limits analytically. When necessary, state thatthe limit does not exist.

a . lim x 2

4 x2+3 e . lim x 1

x3 1

x 1 i . lim

x

6 x2 +2 x+1

5 x2

3 x+4

b . lim x 1

x2 + x+2 x+1 f .

lim x 0

x+1 1 x

j . lim x

x2 + x 24 x3 1

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c . lim x 0

x2 +4 g . lim x 4

( x+3 )2 k . lim x

2 x3

x2 +1

d . lim x 3

3 2 x2 10 h . lim x 2

x5 32 x 2 l .

lim x 2

4 x2

3 x2 +5

!. "iven the piecewise f nction #elow, eval ate the limits as$ed.

a . F ( x)={2 x 3 , x 1 ; lim x 1 F ( x)

b . g ( x)={ x+1 for x

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" ( x)= 900 +18 x0.7

#f the revenue from the sale of s ateboards is given by R( x)= 75 x0.8

. $nd the rate

at which the average pro$t per s ateboard is changing when +0 s ateboards have

been built and sold.c. &t a certain factory, the daily output is # ( $ )= 600 $

1/2 units, where $

denotes the capital investment measured in units of P10,000. he current capitalinvestment isP 6,000,000. /stimate the e7ect that an additional capital investmentof P8,000 will have on the daily output.

d. & manufacturer estimates that when 9 units of a particular commodity are

produced,the total cost will be " ( x)=18

x2 +3 x+98 dollars, and furthermore, that

all x

units will be sold when the price is p( x)= 1

3(75 x)

dollars per unit.

i. ind the marginal cost and the marginal revenue.

ii. :se marginal cost to estimate the cost of producing the ninth unit.

iii. What is the actual cost of producing the ninth unit;

iv. :se marginal revenue to estimate the revenue derived from the sale of the ninthunit.

v. What is the actual revenue derived from the sale of the ninth unit;

e." ( x)= 1

5 x2+4 x+57

is the total cost of producing 9 units of a particular

commodity and p( x)=14

(36 x) is the price at which all 9 units will be sold.

&ssume p(9) and 3(9) are in dollars.

(i) ind the marginal cost and the marginal revenue.

(ii) :se marginal cost to estimate the cost of producing the fourth unit.

(iii) ind the actual cost of producing the fourthunit.

(iv) :se marginal revenue to estimate the revenue derived from the sale of thefourth unit.

(v) ind the actual revenue derived from the sale of the fourth unit.

1/. For each f nction, nd the points on the graph at which the tangentline is hori0ontal. f none exist, state that fact. Find also the intervalswhere the derivative is positive and negative.

a. f ( x)= 3 x2 5 x+4

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b. f ( x)= 0.01 x2 +0.4 x+50

c. f ( x)= x3+ x2 +5 x 1