Each limit represents the derivative of func. at # a. State f and a.

[WK95]

Homework Statement

Each limit below represents the derivative of some function f at some number a. State such an f and a.
$\lim_{x \rightarrow \pi/4} \frac{tan(x) - 1}{x - \pi/4}$

Homework Equations

$f'(x) = \lim_{x \rightarrow 0} \frac{f(a + h) - f(a)}{h}$

The Attempt at a Solution

$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(x) - 1}{x - \frac{\pi}{4}}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(x) - tan(\pi/4)}{x - \frac{\pi}{4}}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(\frac{\pi}{4} +(x - \frac{\pi}{4})) - tan(\frac{\pi}{4})}{x - \frac{\pi}{4}}$
$h = x - \frac{\pi}{4}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(\frac{\pi}{4} + h) - tan(\frac{\pi}{4})}{h}$
$f(x)=tan(x)$
$a=\frac{\pi}{4}$

The definition of the derivative states that x approaches 0. However, in my approach, i get the answer while h approaches pi/4 so I did something incorrectly with my work. However, the end answer seems to be correct. How do I my work to obey the definition of the derivative to solve the problem?

 

[vanhees71]

You should always make a "syntax check" about your formulas! Does what you've written under "Relevant Equations" make any sense?

 

[STEMucator]

Homework Statement

Each limit below represents the derivative of some function f at some number a. State such an f and a.
$\lim_{x \rightarrow \pi/4} \frac{tan(x) - 1}{x - \pi/4}$

Homework Equations

$f'(x) = \lim_{x \rightarrow 0} \frac{f(a + h) - f(a)}{h}$

The Attempt at a Solution

$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(x) - 1}{x - \frac{\pi}{4}}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(x) - tan(\pi/4)}{x - \frac{\pi}{4}}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(\frac{\pi}{4} +(x - \frac{\pi}{4})) - tan(\frac{\pi}{4})}{x - \frac{\pi}{4}}$
$h = x - \frac{\pi}{4}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(\frac{\pi}{4} + h) - tan(\frac{\pi}{4})}{h}$
$f(x)=tan(x)$
$a=\frac{\pi}{4}$

The definition of the derivative states that x approaches 0. However, in my approach, i get the answer while h approaches pi/4 so I did something incorrectly with my work. However, the end answer seems to be correct. How do I my work to obey the definition of the derivative to solve the problem?

Very close. You made a very small error right here though :

$h = x - \frac{\pi}{4}$
$\lim_{x \rightarrow \frac{\pi}{4}} \frac{tan(\frac{\pi}{4} + h) - tan(\frac{\pi}{4})}{h}$

As $x → \frac{π}{4}$ you can observe that $h → 0$ from $h = x - \frac{\pi}{4}$.

This means you should change your limit from $x → \frac{π}{4}$ to $h → 0$.