AnswersPrecalculusLimits, Asymptotes, and End Behavior

Limits, Asymptotes, and End Behavior — Quiz Answers

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Consider f of x is equal to the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 Which statement correctly uses limits to determine the end behavior of f of x ?

A
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 x squared and denominator 1 , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow infinity .
B
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 and denominator x squared , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 0 .
C
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of 1 over 1 , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 1 .
D
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 and denominator x squared , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 7 .
2
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Which statement correctly interprets this limit? limit x to plus or minus infinity of the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1

A
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a horizontal asymptote at y is equal to 7 .
B
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a vertical asymptote at x is equal to 7 .
C
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a horizontal asymptote at y is equal to negative 5 halves .
D
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a vertical asymptote at x is equal to negative 5 halves .
3

Which statement correctly identifies an asymptote of f of x is equal to the fraction with numerator 16 x squared minus 35 and denominator x squared minus 5 using limits?

A
Because limit x to plus or minus infinity of f times x is equal to 7 , the function has a horizontal asymptote at y is equal to 7 .
B
Because limit x to plus or minus infinity of f times x is equal to 16 , the function has a vertical asymptote at x is equal to 16 .
C
Because limit x to plus or minus infinity of f times x is equal to 16 , the function has a horizontal asymptote at y is equal to 16 .
D
Because limit x to plus or minus infinity of f times x is equal to 7 , the function has a vertical asymptote at x is equal to 7 .
4

Which statement describes whether the function f of x is equal to the fraction with numerator 8 times open paren x minus 1 close paren squared and denominator open paren x minus 8 close paren squared has a vertical asymptote at x is equal to 8 ?

A
There is not a vertical asymptote at x is equal to 8 because limit x to 8 of f times x does not exist.
B
There is a vertical asymptote at x is equal to 8 because limit x to 8 raised to the negative power of f times x is equal to infinity and limit x to 8 raised to the plus power of f times x is equal to infinity .
C
There is a vertical asymptote at x is equal to 8 because limit x to infinity of f times x is equal to 8 .
D
There is not a vertical asymptote at x is equal to 8 because limit x to 8 raised to the negative power of f times x and limit x to 8 raised to the plus power of f times x are not opposites.
5

Which function has a horizontal asymptote at y is equal to 3 ?

A
f of x is equal to the fraction with numerator 3 x to the 4th power minus x cubed plus x plus 5 and denominator x to the 5th power minus 4
B
f of x is equal to the fraction with numerator 15 x to the 4th power minus x squared plus 9 and denominator 5 x cubed minus 4
C
f of x is equal to the fraction with numerator 6 x squared minus x plus 4 and denominator 2 x squared minus 1
D
f of x is equal to the fraction with numerator 3 x squared plus 5 x plus 18 and denominator 2 x squared plus 6
6

Consider f of x is equal to the fraction with numerator 8 times open paren x minus 1 close paren and denominator x squared plus 2 x minus 3 Which statements describe the existence of vertical asymptotes at x is equal to negative 3 and x is equal to 1 ?

A
At x is equal to negative 3 , limit x to negative 3 raised to the negative power of f times x is equal to negative infinity and limit x to negative 3 raised to the plus power of f times x is equal to infinity , so there is a vertical asymptote. At x is equal to 1 , limit x to 1 raised to the negative power of f times x is equal to 2 and limit x to 1 raised to the plus power of f times x is equal to 2 ,so there is no vertical asymptote.
B
At x is equal to negative 3 , limit x to negative 3 raised to the negative power of f times x is equal to negative infinity and limit x to negative 3 raised to the plus power of f times x is equal to infinity , so there is a vertical asymptote. At x is equal to 1 , limit x to 1 raised to the negative power of f times x is equal to infinity and limit x to 1 raised to the plus power of f times x is equal to infinity ,so there is also a vertical asymptote.
C
At x is equal to negative 3 , limit x to negative 3 raised to the negative power of f times x is equal to infinity and limit x to negative 3 raised to the plus power of f times x is equal to negative infinity , so there is a vertical asymptote. At x is equal to 1 , limit x to 1 raised to the negative power of f times x is equal to infinity and limit x to 1 raised to the plus power of f times x is equal to infinity ,so there is also a vertical asymptote.
D
At x is equal to negative 3 , limit x to negative 3 raised to the negative power of f times x is equal to infinity and limit x to negative 3 raised to the plus power of f times x is equal to negative infinity , so there is a vertical asymptote. At x is equal to 1 , limit x to 1 raised to the negative power of f times x is equal to 2 and limit x to 1 raised to the plus power of f times x is equal to 2 ,so there is no vertical asymptote.
7

For function limit x to c raised to the negative power of h times x is equal to negative infinity and limit x to c raised to the plus power of h times x is equal to infinity Which statement describes the existence of an asymptote at c ?

A
There must be a horizontal asymptote at y is equal to c .
B
There is no asymptote at c because the one-sided limits are not equivalent.
C
There must be a vertical asymptote at x is equal to c .
D
There is no asymptote at c because the one-sided limits do not exist.
8

Which line represents a horizontal asymptote of f of x is equal to negative the fraction with numerator 5 x squared minus 3 x plus 2 and denominator 10 x squared minus 1 ?

A
y is equal to 2
B
y is equal to negative 0 point 5
C
y is equal to negative 2
D
y is equal to 0 point 5
9

Consider f of x is equal to the fraction with numerator negative 14 x to the 4th power plus 2 x plus 18 and denominator x to the 4th power minus 3 x squared Which statement represents the correct interpretation of this limit? limit x to plus or minus infinity of f times x

A
The limit shows that as x to plus or minus infinity comma f of x to 0 .
B
The limit shows that as x to plus or minus infinity comma f of x to negative infinity .
C
The limit shows that as x to plus or minus infinity comma f of x to negative 14 .
D
The limit shows that as x to plus or minus infinity comma f of x to 14 .
10

Which statement correctly demonstrates using limits to determine a vertical asymptote of g of x is equal to the fraction with numerator 2 times open paren x plus 4 close paren squared and denominator x squared minus 16 ?

A
There is a vertical asymptote at x is equal to negative 4 because limit x to 4 raised to the negative power of g times x is equal to negative infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .
B
There is a vertical asymptote at x is equal to negative 4 because limit x to 4 raised to the negative power of g times x is equal to infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .
C
There is a vertical asymptote at x is equal to 4 because limit x to 4 raised to the negative power of g times x is equal to negative infinity and limit x to 4 raised to the plus power of g times x is equal to infinity .
D
There is a vertical asymptote at x is equal to 4 because limit x to 4 raised to the negative power of g times x is equal to infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .

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