Question

Each limit represents the derivative of some function $f$ at some number $a .$ State such an $f$ and $a$ in each case. $$ \lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4} $$

Name: Problem 34, Chapter 2: Essential Calculus Early Transcendentals
Uploaded: 2020-07-10T14:36:47-08:00
Duration: 1 min 6 s
Channel: Benjamin Schreyer
Description: Each limit represents the derivative of some function f at some number a . State such an f and a in each case. limx →π/ 4 (tanx-1)/(x-π/ 4)

   Each limit represents the derivative of some function
$f$ at some number $a .$ State such an $f$ and $a$ in each case.
$$
\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}
$$

Essential Calculus Early Transcendentals

James Stewart 2nd Edition

Chapter 2, Problem 34

Instant Answer

Solved by Expert Benjamin Schreyer

07/10/2020

Step 1

The function is usually found in the numerator of the limit definition of a derivative. In this case, we have $\tan x - 1$ in the numerator. Show more…

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Each limit represents the derivative of some function $f$ at some number $a .$ State such an $f$ and $a$ in each case. $$ \lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4} $$

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Key Concepts

Derivative

The derivative of a function at a point, commonly defined as f'(a) = lim??? (f(x) ? f(a))/(x ? a), measures the instantaneous rate of change of the function. This concept is central to calculus and is used to understand how functions change at very small scales around a specific input.

Limit Definition of the Derivative

This definition expresses how the behavior of a function as its input approaches a given value can be captured by a limit. By comparing the difference in function values to the corresponding differences in inputs, it formalizes the concept of the slope of a tangent line to the function at that point, providing the foundational idea behind differentiation.

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