183)
Find: lim
x
0
2
x. If the limit does not exist, so state or use the symbol or – if appropriate.
183)
184)
The demand function for a certain product is given by p(x) =50,000
(x+4)2 where p is the price
in dollars and x is the quantity sold. Find lim
x
p(x).
184)
185)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1– 2x+3x2
7 – 8x+ 11x3
185)
186)
Find: lim
x–
x3+ 1
x2+ 2 . If the limit does not exist, so state or use the symbol or – if
appropriate.
186)
187)
An open box is formed by cutting a square piece out of each corner of a 8–inch by 12–inch
piece of metal. If each side of the squares cut out is x inches long, the volume of the box is
given by
V(x) =x(8 – 2x)(12 – 2x).This problem only makes sense when this volume is positive.
Find the values of x for which the volume is positive.
187)
188)
Find the value(s) of x for which f(x) =2x+ 1, if x
1
3, if x< 1 is discontinuous.
188)
189)
The revenue R for a product is given by R(x) = 1750x–x2 where x is the number of units
sold. Graph this function on your graphing calculator in several different viewing
rectangles. Discuss whether you think this function is continuous or not.
189)
190)
The volume of helium in a spherical balloon (in cubic centimeters) as a function of the
radius, r, in centimeters, is given by V(r) =4
3r3. Graph V(r) in the standard viewing
rectangle, –10,10 × – 10,10 and use TRACE to estimate lim
r
1.1 V(r).
190)
191)
If f(x) = 2x+ 7, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
191)
192)
Why can we say that f(x) =x5– 9x4 + 7x3– 5x2– 8x + 1 is continuous on ( , )?
192)
193)
Suppose that the rent for an apartment in a city with rent control is $540 per month. Show
that the rent function R(t) = 540 is continuous at t= 5.
193)
194)
By looking at the graph below, state whether the function f(x) is continuous or
discontinuous at x= – 1.
194)
195)
Find: lim
p
4
p2– 7p+ 12
p2– 3p– 4
195)
196)
Find: lim
x
0–
x – 1
x. If the limit does not exist, so state or use the symbol or – if
appropriate.
196)
197)
If the profit function for a certain business is given by: P(x) = 225x–3x2– 800, use the rule
about the limit of a polynomial function to determine lim
x
20
P(x).
197)
198)
A container manufacturer will make an open box by cutting a 3–inch square from each
corner of a square sheet of aluminum and then turning up the sides. The box is to contain
at least 300 in3. Find the dimensions of the smallest sheet of aluminum that can be used.
198)
199)
Find: lim
h
0
5 +h
x
199)
200)
The rate of change of productivity p (in number of units produced per hour) increases with
time on the job by the function p=50(t2+ 4t)
t2+ 3t+ 20 . Find lim
t
4
p.
200)
201)
The profit function for a product is given by P(x) = 37x+x2– 6300. Determine for what
values of x the profit is positive.
201)
202)
The cost C of producing x units of a certain product is given by C(x) = 10,000 + 50x+4x2.
Use your graphing calculator to explore lim
x
C(x) and discuss what this means.
202)
203)
The profit function for a certain business is given by: P(x) = 225x–3x2– 800. Graph this
function on your graphing calculator and use the evaluation function to determine
lim
x
43.9 P(x) using the rule about the limit of a polynomial function.
203)
204)
Find: lim
x–2
x+ 2
x2– 3
204)
205)
Find: lim
x
4–
3
4 –x. If the limit does not exist, so state or use the symbol or – if
appropriate.
205)
206)
Use the definition of continuity to show f(x) = 2x+ 3 is continuous at x= – 1.
206)
207)
Find: lim 15
x
8
207)
208)
The revenue function for a certain product is given by R(x) = 500x–6x2. Graph this
function in the window 0, 4 ×0, 2000 . Use TRACE to estimate lim
x
3.7
R(x).
208)
209)
Find: lim
t
2
t2+ 2t– 8
t– 2
209)
210)
Find: lim
x
4
2x+ 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
210)
211)
The greatest integer function in mathematics (denoted f(x) =x) is used every day by
cashiers making change for customers. This function tells the amount of paper money for
each amount of change owed. (For example, if the customer is owed $1.25 in change, he
would get $1 in paper money, thus 1.25 = 1). By considering 2.90 , 2.95 , 2.99 , 3.01 ,
and 3.05 , determine lim
x
3x.
211)
212)
Solve: x2(x2– 9) < 0
212)
213)
Find: lim
x
2
x2– 5x+ 2
3x– 4
213)
214)
Find: lim
x
2+
3
2 –x. If the limit does not exist, so state or use the symbol or – if
appropriate.
214)
215)
Find: lim
x
2x2– 4x+ 9
3x2– 8 . If the limit does not exist, so state or use the symbol or – if
appropriate.
215)
216)
Find: lim
x–
2 –x
x2+ 3x. If the limit does not exist, so state or use the symbol or – if
appropriate.
216)
217)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x–
ex–2– 4
5
217)
218)
The cost of purifying water is given by C=10,000
p– 1250 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Find lim
p
0C.
218)
219)
Use the definition of continuity to show that f(x) =x+ 1
x– 1 is continuous at x= 3.
219)
220)
The revenue R for a product is given by R(x) = 2500x–x2 where x is the number of units
sold. Is this function continuous for x= 50?
220)
221)
Find: lim
x
2x2– 4
6x3+2x2. If the limit does not exist, so state or use the symbol or – if
appropriate.
221)
222)
Use the definition of continuity to show that f(x) =5
x – 7 is continuous at x= 6.
222)
223)
Find: lim
x
4x4– 2x2– 3
x2– 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
223)
224)
An open box is formed by cutting a square piece out of each corner of a 12–inch by 10–inch
piece of metal. If each side of the squares cut out is x inches long, the volume of the box is
given by
V(x) =x(12 – 2x)(10 – 2x). This problem only makes sense when this volume is positive.
Find the values of x for which the volume is positive.
224)
225)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =7500(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
225)
226)
Consider the graph of f(x)
Determine the following limits:
(a) lim
x
3+f(x)(d) f(3)
(b) lim
x
3–
f(x)(e) lim
x
f(x)
(c) lim
x
3f(x)(f) lim
x
f(x)
226)
227)
Find: lim
x
4x3– 2
7 –2x3. If the limit does not exist, so state or use the symbol or – if
appropriate.
227)
228)
The length of a material increases as it is heated up according to the equation l= 100 + 2x.
The rate at which the length is increasing is given by: lim
h
0
100 + 2(x+h) – (100 + 2x)
h.
Calculate this limit.
228)
229)
Find: lim
q–3
3q2–q– 5
10
229)
230)
Solve the inequality: x2+ 4
x2+ 1
> 0.
230)
231)
Find: lim (2x2– 4x+ 3)
x
3
231)
232)
The cost of purifying water is given by C=18,000
p– 2000 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Graph this function on
your graphing calculator and determine lim
p
0C.
232)
233)
The yearly sales y of a certain company (in thousands of dollars) is related to the amount
that company spends on advertising, x (in thousands of dollars), according to the equation
y(x) =500x
x+ 20 . Find lim
x
y(x) and discuss what this means to the company.
233)
234)
If the profit function for a certain business is given by: P(x) = 225x–3x2– 800, use the rule
about the limit of a polynomial function to determine lim
x
80
P(x).
234)
235)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
2–
1– 2x
(x2+x– 6)
235)
236)
The yearly sales y of a certain company (in thousands of dollars) is related to the amount
that company spends on advertising, x (in thousands of dollars), according to the equation
y(x) =1000x
x+ 50 . Find lim
x
y(x) and discuss what this means to the company.
236)
237)
Find: lim
h
0
3(x+h) – 3x
h
237)
238)
The cost C of producing x units of a certain product is given by C(x) = 10,000 + 50x+4x2.
Find lim
C(x) and discuss what this means.
238)
239)
Find all points where y=f(x) =2 –x–x2
15 – 2x–x2 is discontinuous.
239)
240)
A television repair technician charges $100 for the first hour of work at your house and $50
for every hour (or fraction thereof) afterwards. The function for what the visit will cost you
is given by:
f(x) =
$100 0 <x
1
$150 1 <x
2
$200 2 <x
3
$250 3 <x
4
(a) Find lim
x
1f(x)
(b) Find lim
x
2.5
f(x)
(c) Find lim
x
f(x)
240)
241)
The cost C of producing x units of a certain product is given by
C(x) = 50,000 + 200x+ 0.3x2. Find lim
x
C(x) and discuss what this means.
241)
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