Prove the quotient theorem using the limit definition?

In summary, the conversation involves a new member of a forum seeking help with proving the quotient theorem using the limit definition. The steps involved in proving the theorem are discussed, and the conversation ends with the member thanking the others for their help.
  • #1
franz32
133
0
Hello guys!

I'm new here! Well, it feels like this forum is cool and interesting!

Can anyone help me here? =)

How do you prove the quotient theorem using the limit definition?
(Given a limit of f of x as x approaches a is A and a limit of g of x as x approaces a is B).
 
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  • #2
[tex]
\begin{equation*}\begin{split}
\lim_{h\rightarrow 0} \frac {\frac {f(x+h)} {g(x+h)} - \frac {f(x)} {g(x)}} {h}
&= \lim_{h\rightarrow 0} \frac {f(x+h)g(x) - f(x)g(x+h)} {hg(x+h)g(x)} \\
&= \lim_{h\rightarrow 0} \frac {f(x+h)g(x) - f(x)g(x) + f(x)g(x) - f(x)g(x+h)} {hg(x+h)g(x)}
\end{split}\end{equation*}
[/tex]
You can take it from here. Be careful. How do you know
[tex]\lim_{h\rightarrow0} g(x+h)=g(x)[/tex]?
 
Last edited:
  • #3
Thank you... =)

Hello. =)

Well, I think I can take it from here. If I have doubts,

I would probably want to clarify it.

Anyway, thank you very much!
 

1. What is the quotient theorem?

The quotient theorem is a mathematical theorem that states the limit of the quotient of two functions is equal to the quotient of their limits, as long as the denominator's limit is not equal to zero.

2. What is the limit definition?

The limit definition is a mathematical concept that describes the behavior of a function as its input approaches a particular value. It is used to formally define the concept of a limit.

3. How is the quotient theorem proven using the limit definition?

The quotient theorem is proven by using the definition of a limit to show that the limit of the quotient of two functions is equal to the quotient of their limits, as long as the denominator's limit is not equal to zero.

4. What is the importance of the quotient theorem in mathematics?

The quotient theorem is an important concept in mathematics because it allows us to evaluate limits of functions that involve division. It also helps us understand the behavior of functions near points where the denominator is equal to zero.

5. Can you provide an example of how to use the quotient theorem to solve a mathematical problem?

Sure, let's say we have the functions f(x) = x^2 and g(x) = x. We want to evaluate the limit of (f(x)/g(x)) as x approaches 2. By using the quotient theorem and the definition of a limit, we can simplify the expression to (2/2) = 1, which is the value of the limit.

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