# Proving Simple Limit Statement: f(x) & f(a+h)

• mscbuck
In summary, the given statement is a proof that the limit of f(x) as x approaches a is equal to the limit of f(a+h) as h approaches 0. This can be shown by defining the left-hand side (LHS) and right-hand side (RHS) of the equation and observing that they are equivalent when h is made to approach 0. This is due to the fact that the limit of x approaching a is equivalent to the limit of x+b approaching a+b, and substituting h = x-a into the equation. However, for a more rigorous proof, one may use the \epsilon,\delta definition of the limit and apply a change of variables.
mscbuck

## Homework Statement

Prove:
"lim f(x) as x-->a is equal to lim f(a+h) as h-->0 (this is really just an exercise in understanding what the terms are)"

## The Attempt at a Solution

Now the easiest way would just to pop in the values and boom, lim f(a) = lim f(a). But I think he is looking for a different reasoning. My attempt was to kind of define the statements like so:

_______

LHS: f can be made to be as close to a limit L as desired by making x sufficiently close to a.

RHS: f can be made to be as close to a limit L as desired by making h sufficiently close to zero.

And in this example, it just so happens when h is made to go to zero we are left with the respective equations L's equal to each other.
______

I don't think I've really "proved" anything though, have I?

Thanks!

x + a - a = x correct? Let's h = x - a. that means:

x = h + a
h = x - a

also you should know this fact, lim x-->a is the smae as x+b-->a+b
For example the limit as x tends to 2, is the same as the limit as x+2 tends to 4.
so x->a is the same as x-a->a-a.

Sub these into: f(x) as x-->a
and you get f(h+a) as x-a -> a-a
which is equivlent to f(h+a) as h-> 0

You're probably expected to go back to the $$\epsilon,\delta$$ definition of the limit and use those techniques.

figured he didn't have to do that since he said
this is really just an exercise in understanding what the terms are

but a change of variables is rigorous and I'm pretty sure if you went back to δ ε you'd just do the same idea there.

## 1. What is a simple limit statement in mathematics?

A simple limit statement is a statement that describes the behavior of a function as the input values get closer and closer to a certain value. It is used to determine the limit of a function at a specific point.

## 2. How do you prove a simple limit statement using f(x) and f(a+h)?

To prove a simple limit statement using f(x) and f(a+h), you must show that the value of the function f(x) approaches a specific value as x approaches a+h. This can be done by using algebraic manipulation or by using the definition of a limit.

## 3. What is the definition of a limit in mathematics?

The definition of a limit in mathematics is a mathematical concept that describes the behavior of a function as the input values approach a certain value. It is the value that the function approaches as the input values get closer and closer to a specific point.

## 4. How do you determine if a limit exists for a function?

A limit exists for a function if the left-sided limit and the right-sided limit are equal at a specific point. This can also be determined by graphing the function and observing if there is a point of discontinuity.

## 5. Can a simple limit statement be proven using other methods besides f(x) and f(a+h)?

Yes, a simple limit statement can also be proven using other methods such as the Squeeze Theorem, the Intermediate Value Theorem, or using the properties of limits. These methods can provide alternative ways to prove a simple limit statement.

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