Find Limit: 1/[x+Δχ] - 1/x as x->0-

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In summary, the conversation is discussing finding the limit of (1/[x+ΔX] - 1/x) / ΔX as x approaches 0 from the left side of the graph. The participants discuss cross multiplication and finding a common denominator to simplify the expression, ultimately resulting in the limit being -1/x^2 as ΔX approaches 0.
  • #1
louie3006
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Homework Statement



find the limit if it exist :
lim (1/[x+Δχ] - 1/x) / Δχ
x->0-





The Attempt at a Solution


lim (1/[x+Δχ] - 1/x) / Δχ => lim -Δχ /(x+ Δx) / Δχ
x->0-
then what ? do i cancel the ΔX in the numerator with that in the denominator ? or what's the next step to solve this problem and did I do any mistake?
 
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  • #2
That limit is undefined as x->0. Are you sure you don't mean delta X ->0? And, if so, sure, you want to cancel the thing that's going to 0. And don't write things like a/b/c without parentheses. It's not clear whether you mean (a/b)/c or a/(b/c). They are different.
 
  • #3
oh yes, I'm sorry , I miss typed it its supposed to be delta x approaching 0 from the left hand side of the graph.
 
  • #4
Ok. Let's write h instead of delta X, ok. You want limit h->0 (1/(x+h)-1/x)/h. I'm not really happy with -h/(x+h)/h for reasons beyond the parentheses.
 
  • #5
You have 1/(x+dx) - 1/x

Cross multiply them, then see what you're left with.

edit: Oh you already did that, the dx cancels and as dx->0 it tends to -1/x^2 no?
 
  • #6
it matches my answer but i wanted to make sure that i did the proper steps
 
  • #7
Gregg said:
You have 1/(x+dx) - 1/x
Cross multiply them, then see what you're left with.
I think I understand what you meant to say, but cross multiplication applies when you have an equation with two rational expressions, such as
a/b = c/d

"Cross multiplication" results in ad = bc, and is equivalent to multiplying both sides of the equation by bd.
 
  • #8
Mark44 said:
I think I understand what you meant to say, but cross multiplication applies when you have an equation with two rational expressions, such as
a/b = c/d

"Cross multiplication" results in ad = bc, and is equivalent to multiplying both sides of the equation by bd.

I mean

x/x(x+dx) - (x+dx)/x(x+dx) to get -dx/(x^2+dx)
 
  • #9
He meant, "get a common denominator and subtract the two fractions."
 

1. What is a limit in mathematics?

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

2. How do you find the limit of a function?

To find the limit of a function, you can use various methods such as direct substitution, factoring, or using properties of limits. In this specific case, we can use the property that the limit of the difference of two functions is equal to the difference of their limits.

3. What does the notation "x -> 0" mean?

The notation "x -> 0" represents the input of a function approaching the value of 0. In other words, it means that we are evaluating the function as the input gets closer and closer to 0.

4. Why is finding limits important in mathematics?

Finding limits is important in mathematics as it helps us understand the behavior of a function and its values as the input changes. It also allows us to solve various mathematical problems and make predictions about the behavior of a system.

5. How do you solve the given limit: 1/[x+Δχ] - 1/x as x->0-

To solve this limit, we can use the property mentioned in question 2 and rewrite the expression as [1/(x+Δχ) - 1/x] = [x/(x(x+Δχ)] - [x+Δχ]/(x(x+Δχ)) = [(x-x-Δχ)/(x(x+Δχ))] = [-Δχ/(x(x+Δχ))]. As x approaches 0, the denominator becomes 0, and the fraction approaches infinity, so the limit does not exist.

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