(c) $$ h(x)=\frac{2}{x} $$ $$ \frac{h(x+h)}{h}=\frac{2}{x+h}= $$

Question

UpStudy AI · Accepted Answer

We start with the function

$$
h(x)=\frac{2}{x}.
$$

The difference quotient is defined by

$$
\frac{h(x+h)-h(x)}{h}.
$$

Step 1: Compute $$ h(x+h) $$:

$$
h(x+h)=\frac{2}{x+h}.
$$

Step 2: Write the expression for the difference quotient:

$$
\frac{h(x+h)-h(x)}{h} = \frac{\frac{2}{x+h} - \frac{2}{x}}{h}.
$$

Step 3: Combine the fractions in the numerator:

$$
\frac{2}{x+h} - \frac{2}{x} = \frac{2x - 2(x+h)}{x(x+h)}.
$$

Simplify the numerator:

$$
2x - 2(x+h) = 2x - 2x - 2h = -2h.
$$

So, the numerator becomes

$$
\frac{-2h}{x(x+h)}.
$$

Step 4: Divide by $$ h $$:

$$
\frac{\frac{-2h}{x(x+h)}}{h} = \frac{-2h}{h \cdot x(x+h)}.
$$

Cancel $$ h $$ (assuming $$ h \neq 0 $$):

$$
\frac{-2}{x(x+h)}.
$$

Thus, the final simplified expression is

$$
-\frac{2}{x(x+h)}.
$$
$$ \frac{h(x+h)}{h} = -\frac{2}{x(x+h)} $$

(c) \( h(x)=\frac{2}{x} \) \( \frac{h(x+h)}{h}=\frac{2}{x+h}= \)