Finding the Tangent Line to a Hyperbola: A Worked Example Using Limits

In summary, the conversation is discussing a worked example from Stewart's Early Transcendentals 6e section 2.7 p. 145. The example involves finding an equation of the tangent line to a hyperbola at a given point, using the equation m = \lim_{h\rightarrow 0}\frac{f(a+h) - f(a)}{h}. The conversation also includes a discussion about the proper use of the distributive law and the importance of recognizing the negative sign in calculations.
  • #1
naele
202
1

Homework Statement



This is a worked example from Stewart's Early Transcendentals 6e section 2.7 p. 145 for anybody curious.

Let [tex]f(x)= \frac{3}{x}[/tex]. Find an equation of the tangent line to the hyperbola at point (3,1).

Homework Equations



[tex]m = \lim_{h\rightarrow 0}\frac{f(a+h) - f(a)}{h}[/tex]

The Attempt at a Solution



His solution goes as such:

1) [tex]m = \lim_{h\rightarrow 0}\frac{f(3+h) - f(3)}{h}= \lim_{h\rightarrow 0}\frac{\frac{3}{3+h} -1}{h}[/tex]
Plug in the point coordinates into the equation and evaluate.

2) [tex]\lim_{h\rightarrow 0}\frac{\frac{3-(3+h)}{3+h}}{h}[/tex]
Consider the 1 as 1/1, cross multiply and multiply through the denominator. The reverse of partial fraction decomposition (recomposition?)

3) [tex]\lim_{h\rightarrow 0}\frac{-h}{h(3+h)}[/tex]
This is where I become confused. Do you have to distribute the negative sign such that 3-(3+h) = 3-3-h = -h?

4) [tex]\lim_{h\rightarrow 0}-\frac{1}{3+h}=-\frac{1}{3}[/tex]
I would have gotten 1/3 instead of -1/3 so I would have made a mistake between steps 2 and 3.
 
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  • #2
Would it help if I wrote it this way?
3 - (3 + h) = 3 + (-1)(3 + h)

Also, if you'd gotten 1/3 you knew you would be wrong, because the graph is clearly going down so the derivative should be negative.
 
Last edited:
  • #3
Yes! But then why did Stewart write it like that? It's so confusing.
 
  • #4
naele said:
Yes! But then why did Stewart write it like that? It's so confusing.
Perhaps Stewart didn't think he had to remind you that -(3-h)= -3+ h. You shouldn't have to ask if you use the distributive law!
 

FAQ: Finding the Tangent Line to a Hyperbola: A Worked Example Using Limits

What is a limit?

A limit is a fundamental concept in mathematics that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value.

Why is it important to understand limits?

Limits are important because they allow us to understand and analyze the behavior of functions, especially in situations where the function is not defined at a particular point. They also help us to evaluate the continuity, differentiability, and other properties of functions.

Can you provide an example of a limit?

Sure, consider the function f(x) = 2x + 1. The limit of this function as x approaches 2 is 5. This means that as x gets closer and closer to 2, the output of the function approaches 5.

How do you solve a limit?

The process of solving a limit involves evaluating the function at the given value and checking if the function approaches a specific value or is undefined at that point. If the function is undefined, additional techniques such as algebraic manipulation or L'Hopital's rule may be used to determine the limit.

Are there any real-life applications of limits?

Yes, limits have many real-life applications, particularly in calculus and physics. For example, limits are used to calculate instantaneous rates of change, determine maximum and minimum values, and analyze the behavior of systems in motion.

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