190)
Find: lim
x
3f(x)
190)
191)
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.
191)
192)
Find the value(s) of x for which f(x) =x+ 2
x3– 4x is discontinuous.
192)
193)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
3 – 5e2–3x
11
193)
194)
Solve: x2
x2+ 1
0
194)
195)
Solve the inequality: x2– 1
x2– 4
< 0.
195)
196)
The profit function for a product is given by P(x) = 35x+x2– 4950. Determine for what
values of x the profit is positive.
196)
197)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
7R(x).
197)
198)
Find: lim
x
4f(x)
198)
199)
The cost of purifying water is given by C=40,000
p– 3750 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Find lim
p
0C.
199)
200)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =8300(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
200)
201)
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) =800x
x+ 10 . Find lim
x
y(x) and discuss what this means to the company.
201)
202)
Find the value(s) of x for which f(x) =x2– 1
x2– 2x– 8 is discontinuous.
202)
203)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =6200(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
203)
204)
Find: lim
x
3+
x2x2– 9. If the limit does not exist, so state or use the symbol or – if
appropriate.
204)
205)
Solve: x– 1
x– 4 0
205)
206)
By looking at the graph below, state whether the function f(x) is continuous or
discontinuous at x= – 1.
206)
207)
Find: lim
x–1+
1
x+ 1 . If the limit does not exist, so state or use the symbol or – if
appropriate.
207)
208)
Let f(x) =
1
3+xif x 0
2
5+xif x< 0
Find all points where this function f(x) is not continuous.
208)
209)
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.
209)
210)
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.
210)
211)
Solve the inequality: xex2–5> 0.
211)
212)
Find: lim 15
x
8
212)
213)
Find the value(s) of x for which f(x) =2x
x2– 3x + 2 is discontinuous.
213)
214)
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= 3.
214)
215)
Solve: (x– 1)(x+ 2)(x– 3) 0
215)
41
216)
Find: lim
x
3+
x2– 6
x+ 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
216)
217)
Find: lim x(x– 1)
x
3
217)
218)
Let f(x) =5, if x> 4
x,if x 4 . For each of the following, find the limit. If the limit does not exist, so
state or use the symbol or – where appropriate. Hint: Sketch the graph of f.
(a) lim
x
4+
f(x)
(b) lim
x
4–
f(x)
(c) lim
x
4
f(x)
(d) lim
x
f(x)
(e) lim
x–
f(x)
218)
219)
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 3.90 , 3.95 , 3.99 , 4.01 ,
and 4.05 , determine lim
x
4x.
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)
Solve the inequality: x2+x
2.
221)
222)
The demand function for a certain product is given by p(x) =1000
(x+10)2 where p is the price
in dollars and x is the quantity sold. Graph this on your graphing calculator in the window
0,100 ×0,10 . Use the TRACE function to find lim
x
p(x). Describe in words what is
happening to the graph and what this means about the demand function.
222)
223)
The revenue R for a product is given by R(x) = 8500x–x2 where x is the number of units
sold. Is this function continuous for x= 20?
223)
224)
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
50
P(x).
224)
225)
Find: lim
x
4–
3
4 –x. If the limit does not exist, so state or use the symbol or – if
appropriate.
225)
226)
A lawyer charges $300 for the first hour of work at your house and $100 for every hour (or
fraction thereof) afterwards. The function for what the visit will cost you is given by:
f(x) =
$300 0 <x
1
$400 1 <x
2
$500 2 <x
3
$600 3 <x
4
(a) Find lim
x
1f(x)
(b) Find lim
x
2.5 f(x)
(c) Find lim
x
f(x)
226)
227)
Find the value(s) of x for which f(x) =
1
x,if x> 1
3x,if x< 1
is discontinuous.
227)
228)
Find: lim
x
2x2– 4x+ 9
3x2– 8 . If the limit does not exist, so state or use the symbol or – if
appropriate.
228)
229)
Solve: x2– 2x– 3 < 0
229)
230)
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)
230)
231)
The cost C of producing x units of a certain product is given by C(x) = 10,000 + 50x+4x2.
Find lim
x
C(x) and discuss what this means.
231)
232)
Use your calculator to complete the table, and use your results to estimate the given limit.
lim
x
3
x2– 9
x– 3
x3.1 3.01 3.001 2.999 2.99 2.9
f(x)
232)
233)
The profit function for a product is given by P(x) = 37x+x2– 6300. Determine for what
values of x the profit is positive.
233)
45
234)
Find: Find lim
x
2
x3– 8
x– 2
234)
235)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x–
ex–2– 4
5
235)
236)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =2x2+ 3x+ 5
236)
237)
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).
237)
238)
Solve: (x–1)2(x+ 4) > 0
238)
239)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1
x– 1
x2– 2x+ 1
239)
240)
Find the value(s) of x for which f(x) =2x+ 1, if x
1
3, if x< 1 is discontinuous.
240)
241)
Solve: x2+ 2x– 3
x– 5
0
241)
46
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