2.4E: Exercises for Section 2.4

Infinite Limits

[T] In exercises 1 - 3, set up a table of values to find the indicated limit. Round to eight significant digits.

1) \(\displaystyle \lim_{x \to 2}\frac{x^2−4}{x^2+x−6}\)

2) \(\displaystyle \lim_{z \to 0}\frac{z−1}{z^2(z+3)}\)

Answer

a. −37.931034; b. −3377.9264; c. −333,777.93; d. −33,337,778; e. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889
\( \displaystyle \lim_{x \to 0}\frac{z−1}{z^2(z+3)}=−∞\)

3) \(\displaystyle \lim_{t \to 0^+}\frac{\cos t}{t}\)

[T] In exercise 4, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?

4) \(\displaystyle \lim_{α \to 0^+} \frac{1}{α}\cos\left(\frac{π}{α}\right)\)

Answer

a. 10.00000; b. 100.00000; c. 1000.0000; d. 10,000.000;
Guess: \(\displaystyle \lim_{α→0^+}\frac{1}{α}\cos\left(\frac{π}{α}\right)=∞\);
Actual: DNE , since the graph shows the function oscillates wildly between values approaching positive infinity and values approaching negative infinity, as the value of \(α\) approaches \(0\) from the positive side.

In exercises 5 - 8, use direct substitution to obtain an undefined expression. Then, simplify the function and determine the limit.

5) \(\displaystyle \lim_{x→−2^−}\frac{2x^2+7x−4}{x^2+x−2}\)

Answer

\(−∞\)

6) \(\displaystyle \lim_{x→−2^+}\frac{2x^2+7x−4}{x^2+x−2}\)

7) \(\displaystyle \lim_{x→1^−}\frac{2x^2+7x−4}{x^2+x−2}\)

Answer

\(−∞\)

8) \(\displaystyle \lim_{x→1^+}\frac{2x^2+7x−4}{x^2+x−2}\)

In exercises 9 - 12, consider the graph of the function\(y=f(x)\) shown here. Which of the statements about \(y=f(x)\) are true and which are false? Explain why a statement is false.

9) \(\displaystyle \lim_{x→10}f(x)=0\)

10) \(\displaystyle \lim_{x→−2^+}f(x)=3\)

Answer

False; \(\displaystyle \lim_{x→−2^+}f(x)=+∞\)

11) \(\displaystyle \lim_{x→−8}f(x)=f(−8)\)

12) \(\displaystyle \lim_{x→6}f(x)=5\)

Answer

False; \(\displaystyle \lim_{x→6}f(x)\) DNE since \(\displaystyle \lim_{x→6^−}f(x)=2\) and \(\displaystyle \lim_{x→6^+}f(x)=5\).

Infinite Limits

In exercises 13 - 17, sketch the graph of a function with the given properties.

13) \(\displaystyle \lim_{x→2}f(x)=1, \quad \lim_{x→4^−}f(x)=3, \quad \lim_{x→4^+}f(x)=6, \quad x=4\) is not defined.

14) \(\displaystyle \lim_{x→−∞}f(x)=0, \quad \lim_{x→−1^−}f(x)=−∞, \quad \lim_{x→−1^+}f(x)=∞,\quad \lim_{x→0}f(x)=f(0), \quad f(0)=1, \quad \lim_{x→∞}f(x)=−∞\)

Answer

Answers may vary

15) \(\displaystyle \lim_{x→−∞}f(x)=2, \quad \lim_{x→3^−}f(x)=−∞, \quad \lim_{x→3^+}f(x)=∞, \quad \lim_{x→∞}f(x)=2, \quad f(0)=-\frac{1}{3}\)

16) \(\displaystyle \lim_{x→−∞}f(x)=2,\quad \lim_{x→−2}f(x)=−∞,\quad \lim_{x→∞}f(x)=2,\quad f(0)=0\)

Answer

Answer may vary

17) \(\displaystyle \lim_{x→−∞}f(x)=0,\quad \lim_{x→−1^−}f(x)=∞,\quad \lim_{x→−1^+}f(x)=−∞, \quad f(0)=−1, \quad \lim_{x→1^−}f(x)=−∞, \quad \lim_{x→1^+}f(x)=∞, \quad \lim_{x→∞}f(x)=0\)

18) Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, \(x\), is shown here. We are mainly interested in the location of the front of the shock, labeled \(X_{SF}\) in the diagram.

a. Evaluate \(\displaystyle \lim_{x→X_{SF}^+}ρ(x)\).

b. Evaluate \(\displaystyle \lim_{x→X_{SF}^−}ρ(x)\).

c. Evaluate \(\displaystyle \lim_{x→X_{SF}}ρ(x)\). Explain the physical meanings behind your answers.

Answer

a. \(ρ_2\) b. \(ρ_1\) c. DNE unless \(ρ_1=ρ_2\). As you approach \(X_{SF}\) from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the “shock” yet and are at a lower density.

Contributors and Attributions

• Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.