Question 1

Find the limit.
-$$\lim _ { x \rightarrow 3 ^ { - } } \frac { 1 } { x ^ { 2 } - 9 }$$

Accepted Answer

A)  $\infty$ 
B)  0 
C)  $- 1$ 
D)  $- \infty$ 
A)  $\infty$ 
B)  0 
C)  $- 1$ 
D)  $- \infty$

Question 2

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { \theta \rightarrow 2 \pi } \tan ( \pi \cos ( \sin \theta ) )$$

Accepted Answer

A)  0; no 
B)  1; yes 
C)  0; yes 
D)  does not exist; no 
A)  0; no 
B)  1; yes 
C)  0; yes 
D)  does not exist; no

Question 3

Answer the question.
-Is $\mathrm { f }$ continuous at $x = 4 ?$
$\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { l l } 
\mathrm { x } ^ { 3 } , & - 2 < \mathrm { x } \leq 0 \
- 2 \mathrm { x } , & 0 \leq \mathrm { x } < 2 \
2 , & 2 < \mathrm { x } \leq 4 \
0 , & \mathrm { x } = 2
\end{array} \right.$

Accepted Answer

A) True 
 B)False

Question 4

Find the intervals on which the function is continuous.
-$$y = \sqrt { x ^ { 2 } - 10 }$$

Accepted Answer

A)  continuous on the interval $[ - \sqrt { 10 } , \sqrt { 10 } ]$ 
B)  continuous on the intervals $( - \infty , - \sqrt { 10 } ]$ and $[ \sqrt { 10 } , \infty )$ 
C)  continuous everywhere 
D)  continuous on the interval $[ \sqrt { 10 } , \infty )$ 
A)  continuous on the interval $[ - \sqrt { 10 } , \sqrt { 10 } ]$ 
B)  continuous on the intervals $( - \infty , - \sqrt { 10 } ]$ and $[ \sqrt { 10 } , \infty )$ 
C)  continuous everywhere 
D)  continuous on the interval $[ \sqrt { 10 } , \infty )$

Question 5

Find numbers a and b, or k, so that f is continuous at every point.
-$f ( x ) = \left\{ \begin{array} { l l } 
1 , & x  3
\end{array} \right.$

Accepted Answer

A)  a = 1, b = -1 
B)  a = -2, b = -7 
C)  a = -2, b = 5 
D)  Impossible 
A)  a = 1, b = -1 
B)  a = -2, b = -7 
C)  a = -2, b = 5 
D)  Impossible

Question 6

Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { x \rightarrow \pi / 2 } \cos \left( \frac { 5 \pi } { 2 } \cos ( \tan x ) \right)$$

Accepted Answer

A)  1; no 
B)  does not exist; no 
C)  1; yes 
D)  does not exist; yes 
A)  1; no 
B)  does not exist; no 
C)  1; yes 
D)  does not exist; yes

Question 7

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$$( x ) = 5 x ^ { 2 } + 4 , x = - 4$$

Accepted Answer

A)  Does not exist 
B)  80 
C)  $- 36$ 
D)  $- 40$ 
A)  Does not exist 
B)  80 
C)  $- 36$ 
D)  $- 40$

Question 8

Find the limit.
-$\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 1$, find $\lim _ { x \rightarrow \theta } \frac { f ( x ) } { x }$

Accepted Answer

A)  2 
B)  1 
C)  0 
D)  Does not exist 
A)  2 
B)  1 
C)  0 
D)  Does not exist

Question 9

Use the graph to evaluate the limit.
- 
$\lim _ { x \rightarrow - 0 } f ( x )$

Accepted Answer

A)  6 
B)  does not exist 
C)  -1 
D)  0 
A)  6 
B)  does not exist 
C)  -1 
D)  0

Question 10

Find the limit.
-$$\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } } { 3 } - \frac { 1 } { x }$$

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\infty$ 
C)  $\frac { 4 } { 3 }$ 
D)  0 
A)  $\frac { 1 } { 3 }$ 
B)  $\infty$ 
C)  $\frac { 4 } { 3 }$ 
D)  0

Question 11

Divide numerator and denominator by the highest power of x in the denominator to find the limit.
-$$\lim _ { x \rightarrow \infty } \frac { 3 x ^ { - 1 } - 2 x ^ { - 3 } } { 5 x ^ { - 2 } + x ^ { - 5 } }$$

Accepted Answer

A)  0 
B)  $\infty$ 
C)  $*$ 
D)  $\frac { 3 } { 5 }$ 
A)  0 
B)  $\infty$ 
C)  $*$ 
D)  $\frac { 3 } { 5 }$

Question 12

Graph the rational function. Include the graphs and equations of the asymptotes.
-$y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } }$

Accepted Answer

A)  asymptotes: $x = 0 , y = 0$
  
B)  asymptote: $y = 0$
  
C)  asymptotes: $x = 0 , y = 0$
  
D)  asymptotes: $x = 0 , y =$
  
A)  asymptotes: $x = 0 , y = 0$
  
B)  asymptote: $y = 0$
  
C)  asymptotes: $x = 0 , y = 0$
  
D)  asymptotes: $x = 0 , y =$

Question 13

Provide an appropriate response.
-Let $\lim _ { x \rightarrow 1 } f ( x ) = 7$ and $\lim _ { x \rightarrow 1 } g ( x ) = - 8$. Find $\lim _ { x \rightarrow 1 } \left[ \frac { - 2 f ( x ) - 4 g ( x ) } { 3 + g ( x ) } \right]$.

Accepted Answer

A)  $= \frac { 18 } { 5 }$ 
B)  $- \frac { 26 } { 3 }$ 
C)  $\frac { 46 } { 5 }$ 
D)  1 
A)  $= \frac { 18 } { 5 }$ 
B)  $- \frac { 26 } { 3 }$ 
C)  $\frac { 46 } { 5 }$ 
D)  1

Question 14

Use the graph to evaluate the limit.
-$\lim _{x-\theta} f(x)$

Accepted Answer

A)  2 
B)  -2 
C)  does not exist 
D)  0 
A)  2 
B)  -2 
C)  does not exist 
D)  0

Question 15

Find the limit if it exists.
-$$\lim _ { x \rightarrow 6 } 5 x ( x + 7 ) ( x - 5 )$$

Accepted Answer

A)  330 
B)  -4290 
C)  -330 
D)  30 
A)  330 
B)  -4290 
C)  -330 
D)  30

Question 16

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$$( x ) = \frac { 2 } { x } , x = - 5$$

Accepted Answer

A)  10 
B)  $- \frac { 2 } { 25 }$ 
C)  Does not exist 
D)  $- \frac { 2 } { 5 }$ 
A)  10 
B)  $- \frac { 2 } { 25 }$ 
C)  Does not exist 
D)  $- \frac { 2 } { 5 }$

Question 17

Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended
function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be
continuously extended from the left or from the right and provide the values of the extended functions at x = 0.
Otherwise write "no continuous extension."
-$$f ( x ) = \frac { \tan x } { x }$$

Accepted Answer

A)  f(0) = 1 only from the right 
B)  No continuous extension 
C)  f(0) = 1 
D)  f(0) = 1 only from the left 
A)  f(0) = 1 only from the right 
B)  No continuous extension 
C)  f(0) = 1 
D)  f(0) = 1 only from the left

Question 18

Find the limit.
-$\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Accepted Answer

A)  does not exist 
B)  0 
C)  29 
D)  15 
A)  does not exist 
B)  0 
C)  29 
D)  15

Question 19

Find the limit, if it exists.
-$$\lim _ { x \rightarrow + } \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 5 x + 4 }$$

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $- \frac { 4 } { 3 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  Does not exist 
A)  $\frac { 2 } { 3 }$ 
B)  $- \frac { 4 } { 3 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  Does not exist

Question 20

Divide numerator and denominator by the highest power of x in the denominator to find the limit.
-$$\lim _  { x \rightarrow \infty } \frac { 3 \sqrt { x } + x ^ { - 1 } } { 4 x - 2 }$$

Accepted Answer

A)  $\infty$ 
B)  0 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
A)  $\infty$ 
B)  0 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 1 } { 4 }$

Find the limit. - $\lim _ { x \rightarrow 3 ^ { - } } \frac { 1 } { x ^ { 2 } - 9 }$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { \theta \rightarrow 2 \pi } \tan ( \pi \cos ( \sin \theta ) )$

Find the intervals on which the function is continuous. - $y = \sqrt { x ^ { 2 } - 10 }$

Find numbers a and b, or k, so that f is continuous at every point. - $f ( x ) = \left\{ \begin{array} { l l } 1 , & x < 2 \\a x + b , & 2 \leq x \leq 3 \\- 1 , & x > 3\end{array} \right.$

Find the limit and determine if the function is continuous at the point being approached. - $\lim _ { x \rightarrow \pi / 2 } \cos \left( \frac { 5 \pi } { 2 } \cos ( \tan x ) \right)$

Find the limit. - $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 1$ , find $\lim _ { x \rightarrow \theta } \frac { f ( x ) } { x }$

Use the graph to evaluate the limit. - $Use the graph to evaluate the limit. - \lim _ { x \rightarrow - 0 } f ( x )$ $\lim _ { x \rightarrow - 0 } f ( x )$

Find the limit. - $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } } { 3 } - \frac { 1 } { x }$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { x \rightarrow \infty } \frac { 3 x ^ { - 1 } - 2 x ^ { - 3 } } { 5 x ^ { - 2 } + x ^ { - 5 } }$

Graph the rational function. Include the graphs and equations of the asymptotes. - $y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } }$ $Graph the rational function. Include the graphs and equations of the asymptotes. - y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } }$

Provide an appropriate response. -Let $\lim _ { x \rightarrow 1 } f ( x ) = 7$ and $\lim _ { x \rightarrow 1 } g ( x ) = - 8$ . Find $\lim _ { x \rightarrow 1 } \left[ \frac { - 2 f ( x ) - 4 g ( x ) } { 3 + g ( x ) } \right]$ .

Use the graph to evaluate the limit. - $\lim _{x-\theta} f(x)$ $Use the graph to evaluate the limit. - \lim _{x-\theta} f(x)$

Find the limit if it exists. - $\lim _ { x \rightarrow 6 } 5 x ( x + 7 ) ( x - 5 )$

Find the limit. - $\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)$

Find the limit, if it exists. - $\lim _ { x \rightarrow + } \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 5 x + 4 }$

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - $\lim _ { x \rightarrow \infty } \frac { 3 \sqrt { x } + x ^ { - 1 } } { 4 x - 2 }$

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Exam 3: Limits and Continuity

Find the limit. - lim⁡x→3−1x2−9\lim _ { x \rightarrow 3 ^ { - } } \frac { 1 } { x ^ { 2 } - 9 }limx→3−​x2−91​

Find the limit and determine if the function is continuous at the point being approached. - lim⁡θ→2πtan⁡(πcos⁡(sin⁡θ))\lim _ { \theta \rightarrow 2 \pi } \tan ( \pi \cos ( \sin \theta ) )limθ→2π​tan(πcos(sinθ))

Find the intervals on which the function is continuous. - y=x2−10y = \sqrt { x ^ { 2 } - 10 }y=x2−10​

Find numbers a and b, or k, so that f is continuous at every point. - f(x)={1,x<2ax+b,2≤x≤3−1,x>3f ( x ) = \left\{ \begin{array} { l l } 1 , & x < 2 \\a x + b , & 2 \leq x \leq 3 \\- 1 , & x > 3\end{array} \right.f(x)=⎩⎨⎧​1,ax+b,−1,​x<22≤x≤3x>3​

Find the limit and determine if the function is continuous at the point being approached. - lim⁡x→π/2cos⁡(5π2cos⁡(tan⁡x))\lim _ { x \rightarrow \pi / 2 } \cos \left( \frac { 5 \pi } { 2 } \cos ( \tan x ) \right)limx→π/2​cos(25π​cos(tanx))

Find the limit. - lim⁡x→0f(x)x2=1\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = 1limx→0​x2f(x)​=1 , find lim⁡x→θf(x)x\lim _ { x \rightarrow \theta } \frac { f ( x ) } { x }limx→θ​xf(x)​

Use the graph to evaluate the limit. - lim⁡x→−0f(x)\lim _ { x \rightarrow - 0 } f ( x )limx→−0​f(x)

Find the limit. - lim⁡x→1x23−1x\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } } { 3 } - \frac { 1 } { x }limx→1​3x2​−x1​

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - lim⁡x→∞3x−1−2x−35x−2+x−5\lim _ { x \rightarrow \infty } \frac { 3 x ^ { - 1 } - 2 x ^ { - 3 } } { 5 x ^ { - 2 } + x ^ { - 5 } }limx→∞​5x−2+x−53x−1−2x−3​

Graph the rational function. Include the graphs and equations of the asymptotes. - y=x2+1x3y = \frac { x ^ { 2 } + 1 } { x ^ { 3 } }y=x3x2+1​

Use the graph to evaluate the limit. - lim⁡x−θf(x)\lim _{x-\theta} f(x)limx−θ​f(x)

Find the limit if it exists. - lim⁡x→65x(x+7)(x−5)\lim _ { x \rightarrow 6 } 5 x ( x + 7 ) ( x - 5 )limx→6​5x(x+7)(x−5)

Find the limit. - lim⁡x→2(x3+5x2−7x+1)\lim _ { x \rightarrow 2 } \left( x ^ { 3 } + 5 x ^ { 2 } - 7 x + 1 \right)limx→2​(x3+5x2−7x+1)

Find the limit, if it exists. - lim⁡x→+x2−4x+3x2−5x+4\lim _ { x \rightarrow + } \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 5 x + 4 }limx→+​x2−5x+4x2−4x+3​

Divide numerator and denominator by the highest power of x in the denominator to find the limit. - lim⁡x→∞3x+x−14x−2\lim _ { x \rightarrow \infty } \frac { 3 \sqrt { x } + x ^ { - 1 } } { 4 x - 2 }limx→∞​4x−23x​+x−1​