What is the Limit of the Difference Quotient for f(x) = 2/x as h Approaches 0?

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In summary, to find the limit of (f(x+h) - f(x)) / h as h approaches 0, where f(x) = 2/x and x = -4, use the butterfly method to simplify and then apply the rule for dividing fractions.
  • #1
Chas3down
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Homework Statement



(f(x+h) - f(x)) / h

f(x) = 2/x
x = -4

As h approaches 0

Homework Equations


N/A


The Attempt at a Solution


(2/(-4 + h) + 1/2) / h


Don't know where to go from there though, not sure how to simplify.
 
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  • #2
Chas3down said:

Homework Statement



(f(x+h) - f(x)) / h

f(x) = 2/x
x = -4

As h approaches 0

Homework Equations


N/A

The Attempt at a Solution


(2/(-4 + h) + 1/2) / h Don't know where to go from there though, not sure how to simplify.

So you want to compute this :

##lim_{h→0} \frac{f(x+h) - f(x)}{h}## when ##f(x) = \frac{2}{x}## and ##x=-4##.

My advice is leave the x=-4 until the very end in these types of problems and just work with this :

##lim_{h→0} \frac{\frac{2}{x+h} - \frac{2}{x}}{h}##

Find a common denominator for the numerator and simplify it, then apply this rule :

##\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}##.

You'll be able to find the limit easily afterwards.
 
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  • #3
In response to what Zondrina said...

when simplifying ##lim_{h→0} \frac{\frac{2}{x+h} - \frac{2}{x}}{h}##,
I use a method called the butterfly method...

Just cross multiply the denominator of the left fraction with the numerator of the right fraction and the denominator of the right fraction with the numerator with the left fraction and finally multiply the denominators of both fractions to get this...

##\frac{\frac{2x-2(x+h)}{(x+h)x}}{h}##

Then use the rule suggested by Zondrina with the h: ##\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}##
 
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  • #4
Got it, thanks a lot guys!
 

1. What is the purpose of the limit (f(x+h) - f(x)) / h?

The limit (f(x+h) - f(x)) / h is used to calculate the instantaneous rate of change of a function at a specific point. It is also known as the derivative of the function at that point.

2. How is the limit (f(x+h) - f(x)) / h calculated?

The limit (f(x+h) - f(x)) / h is calculated by taking the limit as h approaches 0. This means that the value of h gets closer and closer to 0, and the resulting value is the instantaneous rate of change at that specific point.

3. Can the limit (f(x+h) - f(x)) / h be used to find the slope of a curve?

Yes, the limit (f(x+h) - f(x)) / h is used to find the slope of a curve at a specific point. This slope is also known as the derivative of the function at that point.

4. What is the significance of the limit (f(x+h) - f(x)) / h in calculus?

The limit (f(x+h) - f(x)) / h is a fundamental concept in calculus as it is used to calculate derivatives, which are essential in determining the rate of change of a function. It is also used to find the slope of a curve, which is crucial in analyzing the behavior of functions.

5. Are there any limitations to using the limit (f(x+h) - f(x)) / h?

One limitation of using the limit (f(x+h) - f(x)) / h is that it can only be used for functions that are continuous at the point of interest. If the function has a discontinuity or a sharp turn at that point, the limit cannot be calculated. Additionally, the limit does not exist for some functions, making it impossible to calculate the derivative at that point.

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