Try It Yourself

Exercise 1

Evaluate the limit

If \(f(x) = -3\), find \(\lim\limits_{x\to4^{+}}f(x)\).

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Answer: \(-3\)

Exercise 2

Evaluate the limit

Evaluate \(\lim\limits_{x\to 0} \dfrac{2x^3 -4x^2+3x-6}{2x-4}\).

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Answer: \(3/2\)

Exercise 3

Evaluate the limit

Evaluate \(\displaystyle\lim_{x\to 1^-}\frac{x^2-4}{x-1}\).

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Answer: DNE (\(+\infty\))

Exercise 4

Evaluate the limit

Evaluate \(\displaystyle\lim_{x\to 3}\frac{x-2}{(x-3)^3}\).

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Answer: DNE

Exercise 5

Evaluate the limit

If \(f(x) = \dfrac{x^4 - 1}{x-1}\), find \(\lim\limits_{x\to1} f(x)\).

Hint: Factor \(x^4-1\) by treating it as a difference of squares, \((x^2)^2 - 1^2\).

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Answer: \(4\)

Exercise 6

Evaluate the limit

Find \(\displaystyle\lim_{x\to 1} g(x)\) and \(\displaystyle\lim_{x\to 2} g(x)\), where

\[\begin{equation*} g(x)= \begin{cases} x^3+5 & \hbox{ if $x<1$} \\ 7-x^2 & \hbox{ if $1<x\leq 2$} \\ 5x-3 & \hbox{ if $x>2$} \\ \end{cases} \end{equation*}\]

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Answer: \(6\), DNE

Exercise 7

Evaluate the limit at infinity

Evaluate \(\displaystyle\lim_{x\to \infty} \frac{x^2-x+3}{2x^2+1}\).

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Answer: \(1/2\)

Exercise 8

Evaluate the limit at infinity

Evaluate \(\displaystyle\lim_{x\to \infty} \frac{7x^4+2x+8}{x^5+1}\).

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Answer: \(0\)

Exercise 9

Evaluate the limit at infinity

Evaluate \(\displaystyle\lim_{x\to \infty} \frac{x^9+1}{2x^3+2}\).

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Answer: DNE

Exercise 10

Average cost at infinity

Penn State Learning produces a line of Math Genius robots. In order to produce them, PSL purchases a factory that costs $35,000 up front. Each Math Genius robot costs $70 to produce. If PSL produces an infinite number of robots, what is the average cost per robot? What can we conclude from this answer?

Hint: Average Cost = \(\frac{\textsf{Total Cost}}{\textsf{Number of Units}}\)

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Answer: Average cost is $70.

Exercise 11

Find the discontinuities

Find all values of \(x\) where \(f(x)= \dfrac{x+2}{x^3-4x}\) is discontinuous.

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Answer: \(-2,0,2\)

Exercise 12

Continuous at a point

Determine whether or not \(f(x)\) is continuous at \(x=3\).

\[\begin{split} f(x) = \begin{cases} \dfrac{x^2-9}{x^2+2x-15} & \hbox{ if $x \neq 3$}\\ 7 & \hbox{ if $x = 3$} \end{cases} \end{split}\]

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Answer: \(f(x)\) is not continuous at \(x=3\).

Exercise 13

Continuity

Find values of \(x\) where \(f(x)\) is continuous

\[\begin{split} f(x) = \begin{cases} 6-2x & \hbox{ if $x<3$}\\ x^2-5x+5 & \hbox{ if $x \geq 3$} \end{cases}. \end{split}\]

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Answer: everywhere except \(x = 3\)

Exercise 14

Continuity

Find the value for \(k\) that makes \(f(x)\) continuous at \(x=2\).

\[\begin{split} f(x) = \begin{cases} kx^2+5 & \hbox{ if $x<2$}\\ 21-kx -x^2 & \hbox{ if $x \geq 2$} \end{cases} \end{split}\]

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Answer: \(k = 2\)

Exercise 15

Continuity

Find the value for \(k\) that makes \(f(x)\) continuous at \(x=3\).

\[\begin{split} f(x) = \begin{cases} \dfrac{x^2-9}{x-3} & \hbox{ if $x \neq 3$}\\ kx-18 & \hbox{ if $x = 3$} \end{cases} \end{split}\]

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Answer: \(k = 8\)