# 2.3E: Exercises for Section 2.3

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## Infinite Limits

[T] In exercises 1 - 2, set up a table of values to find the indicated limit. Round to eight significant digits.

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

$$z$$ $$\frac{z−1}{z^2(z+3)}$$ $$z$$ $$\frac{z−1}{z^2(z+3)}$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.
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
$$\displaystyle \lim_{x \to 0}\frac{z−1}{z^2(z+3)}=−∞$$

2) $$\displaystyle \lim_{t \to 0^+}\frac{\cos t}{t}$$

$$t$$ $$\frac{\cos t}{t}$$
0.1 a.
0.01 b.
0.001 c.
0.0001 d.

[T] In exercises 3 - 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?

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

$$a$$ $$\frac{1}{α}\cos\left(\frac{π}{α}\right)$$
0.1 a.
0.01 b.
0.001 c.
0.0001 d.

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.

4) $$\displaystyle \lim_{θ \to 0}\sin\left(\frac{π}{θ}\right)$$

$$θ$$ $$\sin\left(\frac{π}{θ}\right)$$ $$θ$$ $$\sin\left(\frac{π}{θ}\right)$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.

In exercises 5 - 8, 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.

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

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

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

7) $$\displaystyle \lim_{x→6}f(x)=5$$

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

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

In each exercise, sketch the graph of a function with the given properties.

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

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

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

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

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