Chapter 2: Differentiation

Section 2.2: Precise Definition of the Derivative

Example 2.2.4

$$ Evaluate $\underset{z\to 2}{lim}\frac{\sqrt{z^2\+1}-\sqrt{5}}{z-2}$. $$ Solution $$ •  Control-drag the expression to be evaluated. Context Panel: Evaluate and Display Inline $\underset{z\to 2}{lim}\frac{\sqrt{z^2\+1}-\sqrt{5}}{z-2}$ = $\frac{2}{5}\sqrt{5}$$$ Stepwise solution Rationalize the numerator   $\frac{\sqrt{z^2\+1}-\sqrt{5}}{z-2}$ $\=\frac{\sqrt{z^2\+1}-\sqrt{5}}{z-2}\cdot \frac{\sqrt{z^2\+1}\+\sqrt{5}}{\sqrt{z^2\+1}\+\sqrt{5}}$ $$ $\=\frac{z^2\+1-5}{\left(z-2\right) \left(\sqrt{z^2\+1}\+\sqrt{5}\right)}$ $$ $\=\frac{\left(z\+2\right) \left(z-2\right)}{\left(z-2\right) \left(\sqrt{z^2\+1}\+\sqrt{5}\right)}$ $$ $\=\frac{z\+2}{\sqrt{z^2\+1}\+\sqrt{5}}$   Evaluate at $z\=2$ to get $\frac{4}{2\sqrt{5}}\=2\/\sqrt{5}$, which Maple insists on writing as $\frac{2}{5}\sqrt{5}$. Relate to differentiation •  If $f\left(x\right)\=\sqrt{x^2\+1}$, the given limit is the limit of the difference quotient $\frac{f\left(z\right)-f\left(2\right)}{z-2}$, where $z\to 2$. This is $f\prime \left(2\right)$. •  Control-drag $f\left(x\right)\=\dots$  Context Panel: Assign Function $f\left(x\right)\=\sqrt{x^2\+1}$$\stackrel{\text{assign as function}}{\to }$$f$ •  Write $f\prime \left(2\right)$ Context Panel: Evaluate and Display Inline $f\prime \left(2\right)$ = $\frac{2}{5}\sqrt{5}$$$ $$

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