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

## 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
Gold Member
2021 Award
You should always make a "syntax check" about your formulas! Does what you've written under "Relevant Equations" make any sense?

STEMucator
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## 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##.

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