A. Suppose that f(x) is differentiable for all x in [0,1]...

a. Suppose that $f(x)$ is differentiable for all $x$ in $[0,1]$ and that $f(0)=0 .$ Define sequence $\left\{a_{n}\right\}$ by the rule $a_{n}=n f(1 / n)$ . Show that $\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .$ Use the result in part (a) to find the limits of the following sequences $\left\{a_{n}\right\} .$ $$\begin{aligned} \text { b. } a_{n} &=n \tan ^{-1} \frac{1}{n} & \text { c. } a_{n}=n\left(e^{1 / n}-1\right) \\ \text { d. } a_{n} &=n \ln \left(1+\frac{2}{n}\right) \end{aligned}$$

a. Suppose that $f(x)$ is differentiable for all $x$ in $[0,1]$ and that
$f(0)=0 .$ Define sequence $\left\{a_{n}\right\}$ by the rule $a_{n}=n f(1 / n)$ .
Show that $\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .$ Use the result in part (a) to find
the limits of the following sequences $\left\{a_{n}\right\} .$
$$\begin{aligned} \text { b. } a_{n} &=n \tan ^{-1} \frac{1}{n} & \text { c. } a_{n}=n\left(e^{1 / n}-1\right) \\ \text { d. } a_{n} &=n \ln \left(1+\frac{2}{n}\right) \end{aligned}$$

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Key Concepts

Derivative as the Limit of the Difference Quotient

The derivative at a point, particularly at x = 0 in this context, is defined as the limit of the ratio f(h)/h as h approaches 0. This principle is used to equate the limit of the sequence n f(1/n) with f?'(0), since replacing h with 1/n makes the sequence approach the definition of the derivative at 0.

Linear Approximation and First-Order Taylor Expansion

When a function is differentiable at a point, it can be approximated by its tangent line near that point. Thus, for small x, f(x) ? f(0) + f?'(0)x. This approximation underlies the result that n f(1/n) will tend to f?'(0) as n tends to infinity, since f(0) is given to be 0.

Differentiation of Elementary Functions

Understanding how to differentiate common functions such as the arctan, exponential, and natural logarithm functions is essential. Their derivatives at specific points (here at x = 0) provide a direct way to compute the limits of sequences that involve scaled function values and illustrate the practical application of derivative concepts in determining limits.

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