Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 31

$\displaystyle \lim_{h \to 0} \frac{(1 + h)^{10} - 1}{h}$ represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in this case.

Using the limit equation for derivatives

$\lim\limits_{h \to 0} \displaystyle \frac{f(a + h) - f(a)}{h}$

We can see that $f(a + h) = (1 + h)^{10}$ so $f(x) = x^{10}$

and $a = 1$. To check value of $a$, we have


$
\begin{equation}
\begin{aligned}

f(a) =& a^{10} && \\
\\
f(1) =& (1)^{10} && \\
\\
f(a) =& 1 \qquad \text{Just like the 1 on the right side of the numerator}\\
\\
\text{Therefore,}\\
\\
f(x) =& x^{10} \text{ and } a = 1


\end{aligned}
\end{equation}
$