Question

practice exercises
21 - 44. determining limits analytically determine the following limits.

1. a. $lim_{x

ightarrow2^{+}}\frac{1}{x - 2}$ b. $lim_{x
ightarrow2^{-}}\frac{1}{x - 2}$ c. $lim_{x
ightarrow2}\frac{1}{x - 2}$

1. a. $lim_{x

ightarrow3^{+}}\frac{2}{(x - 3)^{3}}$ b. $lim_{x
ightarrow3^{-}}\frac{2}{(x - 3)^{3}}$ c. $lim_{x
ightarrow3}\frac{2}{(x - 3)^{3}}$

1. a. $lim_{x

ightarrow4^{+}}\frac{x - 5}{(x - 4)^{2}}$ b. $lim_{x
ightarrow4^{-}}\frac{x - 5}{(x - 4)^{2}}$ c. $lim_{x
ightarrow4}\frac{x - 5}{(x - 4)^{2}}$

1. a. $lim_{x

ightarrow1^{+}}\frac{x}{|x - 1|}$ b. $lim_{x
ightarrow1^{-}}\frac{x}{|x - 1|}$ c. $lim_{x
ightarrow1}\frac{x}{|x - 1|}$

1. a. $lim_{z

ightarrow3^{+}}\frac{(z - 1)(z - 2)}{(z - 3)}$ b. $lim_{z
ightarrow3^{-}}\frac{(z - 1)(z - 2)}{(z - 3)}$
c. $lim_{z
ightarrow3}\frac{(z - 1)(z - 2)}{(z - 3)}$

Explanation:

21.

a.

Step1: Analyze right - hand limit

As \(x\to2^{+}\), \(x - 2\to0^{+}\). Let \(t=x - 2\), when \(x\to2^{+}\), \(t\to0^{+}\), and \(\lim_{x\to2^{+}}\frac{1}{x - 2}=\lim_{t\to0^{+}}\frac{1}{t}=+\infty\).

Step1: Analyze left - hand limit

As \(x\to2^{-}\), \(x - 2\to0^{-}\). Let \(t=x - 2\), when \(x\to2^{-}\), \(t\to0^{-}\), and \(\lim_{x\to2^{-}}\frac{1}{x - 2}=\lim_{t\to0^{-}}\frac{1}{t}=-\infty\).

Step1: Check one - sided limits

Since \(\lim_{x\to2^{+}}\frac{1}{x - 2}=+\infty\) and \(\lim_{x\to2^{-}}\frac{1}{x - 2}=-\infty\), the two - sided limit does not exist.

Answer:

\(+\infty\)

b.