Thread: Limits using definition

Can you tell me how to find a limit using the definition only?
For example
Limit of 1/(5x-3) when x tends to 2.

I do know how to prove the above limit is 1/7(if the value 1/7 is given) using the definition. But i do not know how to arrive at the value using the definition of limits

Thanks

Originally Posted by Dili

Can you tell me how to find a limit using the definition only?
For example
Limit of 1/(5x-3) when x tends to 2.

I do know how to prove the above limit is 1/7(if the value 1/7 is given) using the definition. But i do not know how to arrive at the value using the definition of limits

Thanks

Is this what you are looking for?


and


Since the limit exists on both sides and is the same we may unambigously state that


If this isn't what you are looking for, I'm not sure what to say. The point x = 2 is in the domain of the expression, and as such we can usually say that the limit is just the value of the expression with x = 2. (Because of the argument above.)

-Dan

Oh! Unless you mean you need an proof? (I'd have to look that one up myself to make sure I've got the definitions right.)

-Dan

This is just an exercise in algebra.
Note that: .

If then

Here it is. If then choose , then if we have .

QED

Yes, by definition i meant the epsilon delta proof.
But I do know how to prove lim 1/(5x-3) is 1/7 when x tends to 2.(the way Plato proved)
The problem is we were given the limit, just to find the limit of 1/(5x-3) when x tends to 2 using the definition. We were not given the value 1/7, we have to arrive at it
If anyone can help i would be really grateful

Thanks

Originally Posted by Dili

The problem is we were given the limit, just to find the limit of 1/(5x-3) when x tends to 2 using the definition. We were not given the value 1/7, we have to arrive at it.

I think that you have misunderstood the question.
There really is no applying the method to find the limit.
We use the definition to prove that 1/7 is the limit once we have found it by usual means.