Find the one-sided limits of the following functions as `x to 0`
(a) `f(x)=(1)/(2-2^(1//x))`
(b) `f(x)=e^(1//x)`.
(c ) `f(x)=(|sin x|)/(x)`

Answer

To find the one-sided limits of the given functions as \( x \) approaches \( 0 \), we will analyze each function separately.

### (a) \( f(x) = \frac{1}{2 - 2^{\frac{1}{x}}} \)

**Step 1: Find the left-hand limit as \( x \to 0^- \)**

When \( x \) approaches \( 0 \) from the left, \( \frac{1}{x} \) approaches \( -\infty \). Therefore, \( 2^{\frac{1}{x}} \) approaches \( 0 \).

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{1}{2 - 2^{\frac{1}{x}}} = \frac{1}{2 - 0} = \frac{1}{2} \]

**Step 2: Find the right-hand limit as \( x \to 0^+ \)**

When \( x \) approaches \( 0 \) from the right, \( \frac{1}{x} \) approaches \( +\infty \). Therefore, \( 2^{\frac{1}{x}} \) approaches \( +\infty \).

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{1}{2 - 2^{\frac{1}{x}}} = \frac{1}{2 - \infty} = 0 \]

### Summary for (a):
- Left-hand limit: \( \frac{1}{2} \)
- Right-hand limit: \( 0 \)

### (b) \( f(x) = e^{\frac{1}{x}} \)

**Step 1: Find the left-hand limit as \( x \to 0^- \)**

When \( x \) approaches \( 0 \) from the left, \( \frac{1}{x} \) approaches \( -\infty \).

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^{\frac{1}{x}} = e^{-\infty} = 0 \]

**Step 2: Find the right-hand limit as \( x \to 0^+ \)**

When \( x \) approaches \( 0 \) from the right, \( \frac{1}{x} \) approaches \( +\infty \).

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} e^{\frac{1}{x}} = e^{+\infty} = \infty \]

### Summary for (b):
- Left-hand limit: \( 0 \)
- Right-hand limit: \( \infty \)

### (c) \( f(x) = \frac{|\sin x|}{x} \)

**Step 1: Find the left-hand limit as \( x \to 0^- \)**

As \( x \) approaches \( 0 \) from the left, \( |\sin x| \) approaches \( 0 \) and \( x \) approaches \( 0 \) (negative).

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{|\sin x|}{x} = \lim_{x \to 0^-} \frac{-\sin(-x)}{-x} = \lim_{x \to 0^-} \frac{\sin(-x)}{-x} = \lim_{h \to 0} \frac{\sin h}{h} = 1 \]

**Step 2: Find the right-hand limit as \( x \to 0^+ \)**

As \( x \) approaches \( 0 \) from the right, \( |\sin x| \) approaches \( 0 \) and \( x \) approaches \( 0 \) (positive).

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{|\sin x|}{x} = \lim_{x \to 0^+} \frac{\sin x}{x} = 1 \]

### Summary for (c):
- Left-hand limit: \( 1 \)
- Right-hand limit: \( 1 \)

### Final Results:
- (a) Left: \( \frac{1}{2} \), Right: \( 0 \)
- (b) Left: \( 0 \), Right: \( \infty \)
- (c) Left: \( 1 \), Right: \( 1 \)