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

In summary, the problem asks to find a function f and a number a such that the given limit is equal to the derivative of f at a, according to the definition of derivative. The solution involves manipulating the given limit by using the properties of the tangent function and setting a variable h equal to x - pi/4. However, the final answer should be obtained as h approaches 0, not as x approaches pi/4.
  • #1
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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?
 
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  • #2
You should always make a "syntax check" about your formulas! Does what you've written under "Relevant Equations" make any sense?
 
  • #3
WK95 said:

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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1. What is a limit?

A limit in calculus refers to the value that a function approaches as the input variable gets closer and closer to a particular value. It is used to describe the behavior of a function near a specific point.

2. How is a limit calculated?

A limit is calculated by plugging in values that are very close to the given point into the function. The closer the values are to the given point, the more accurate the limit will be. This process is known as "evaluating the limit".

3. What does it mean for a limit to represent the derivative of a function at a specific point?

This means that the slope of the tangent line to the function at that point is equal to the value of the limit. In other words, the derivative of a function at a point is the rate of change of the function at that point.

4. Can a limit represent the derivative of a function at a point if the function is not continuous at that point?

No, for a limit to represent the derivative of a function at a point, the function must be continuous at that point. A function is continuous if it has no breaks, holes, or jumps in its graph at that point.

5. How can limits and derivatives be used in real-world applications?

Limits and derivatives are commonly used in physics, engineering, and economics to model and understand the behavior of various systems. For example, in physics, derivatives are used to calculate velocity and acceleration, while in economics, derivatives are used to analyze supply and demand curves.

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