Textbook 2 Videos $$ [+] $$ Evaluate the limit: $$ \lim _{h \rightarrow 0} \frac{(1+h)^{2}-1}{h} $$ Question Help: Video $$ \square $$ Message instructor Submit Question Jump to Answer

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To evaluate the limit:

$$
\lim_{h \rightarrow 0} \frac{(1+h)^{2} - 1}{h}
$$

let's follow the step-by-step process.

### Step 1: Expand the Numerator

First, expand $$(1 + h)^2$$:

$$
(1 + h)^2 = 1 + 2h + h^2
$$

Now, subtract 1 from the expanded form:

$$
(1 + 2h + h^2) - 1 = 2h + h^2
$$

So, the expression becomes:

$$
\frac{2h + h^2}{h}
$$

### Step 2: Simplify the Expression

Factor $$h$$ out of the numerator:

$$
\frac{h(2 + h)}{h}
$$

Now, cancel out the $$h$$ in the numerator and denominator (assuming $$h \neq 0$$):

$$
2 + h
$$

### Step 3: Evaluate the Limit

Now, take the limit as $$h$$ approaches 0:

$$
\lim_{h \rightarrow 0} (2 + h) = 2 + 0 = 2
$$

### Conclusion

The limit is:

$$
2
$$
The limit is 2.

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