Question about finding the limit of a piecewise function of a piecewise function?

What is the lim as x--->2 of g(f(x)) with reference to this diagram:

ImageShack® - Online Photo and Video Hosting

My reasoning thus far is that there is no limit because as x approaches 2 in the first graph, f(x) approaches 0. But in the second graph, of g(x), the function is not continuous at x = 0.

Is it DNE?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** What is the lim as x--->2 of g(f(x)) with reference to this diagram:

ImageShack® - Online Photo and Video Hosting
My reasoning thus far is that there is no limit because as x approaches 2 in the first graph, f(x) approaches 0. But in the second graph, of g(x), the function is not continuous at x = 0.

Is it DNE?

does not exist. *DNE*?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** why does it not exist?

It may exist.

What is $\displaystyle \lim _{x \to 0^ + } g(x)~?$

Re: Question about finding the limit of a piecewise function of a piecewise function?

That is 1, from the diagram I provided, but the limit does not exist because as x approaches 0 from the left, the limit of g(x) is 3. Is this correct?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** That is 1, from the diagram I provided, but the limit does not exist because as x approaches 0 from the left, the limit of g(x) is 3. Is this correct?

But the problem is $\displaystyle \lim _{x \to 2} g\left( {f(x)} \right)$.

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**Plato** But the problem is $\displaystyle \lim _{x \to 2} g\left( {f(x)} \right)$.

Right, the problem does not specify whether x is approaching 2 from the left or the right side. On the graph of f(x) it approaches 0 from both sides, but if we approach zero on the graph of g(x), g(x) is different depending on whether we approach from the left or right side.

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** On the graph of f(x) it approaches 0 from both sides, but if we approach zero on the graph of g(x), g(x) is different depending on whether we approach from the left or right side.

Is that true? Look again.

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**Plato** Is that true? Look again.

Sorry, I meant to say that on the graph of f(x) as **x** approaches 2 from both sides, f(x) approaches zero. However, on the graph of g(x) you cannot approach 0 from both sides with the limit being the same. Is the correct way of interpreting the question to check the value that f(x) approaches as x approaches 2 on the graph of f(x) and then take that value in order to examine what value(s) g(x) assumes?

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** Is the correct way of interpreting the question to check the value that f(x) approaches as x approaches 2 on the graph of f(x) and then take that value in order to examine what value(s) g(x) assumes?

That is exactly the point.

Here is a bit of notation. $\displaystyle x\approx a$ means that $\displaystyle x$ is very very *close to* $\displaystyle a$ BUT $\displaystyle x\ne a$. That is the concept of limit.

If this case $\displaystyle x\approx 2$ gives $\displaystyle f(x)\approx 0$ but moreover $\displaystyle f(x)>0$ so what is true of $\displaystyle g(f(x))~?$

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**Plato** That is exactly the point.

Here is a bit of notation. $\displaystyle x\approx a$ means that $\displaystyle x$ is very very *close to* $\displaystyle a$ BUT $\displaystyle x\ne a$. That is the concept of limit.

If this case $\displaystyle x\approx 2$ gives $\displaystyle f(x)\approx 0$ but moreover $\displaystyle f(x)>0$ so what is true of $\displaystyle g(f(x))~?$

Thanks a lot! I got the answer. I never looked at it from that perspective, but since f(x) > 0 in from both the left and right side, we must approach the value g(x) thinking that the x-axis is actually the f(x) axis, meaning that we approaching f(x) = 0 from its positive side.

Re: Question about finding the limit of a piecewise function of a piecewise function?

Quote:

Originally Posted by

**lm1988** Thanks a lot! I got the answer. I never looked at it from that perspective, but since f(x) > 0 in from both the left and right side, we must approach the value g(x) thinking that the x-axis is actually the f(x) axis, meaning that we approaching f(x) = 0 from its positive side.

Exactly

$\displaystyle \lim _{x \to 2} f(x) = 0\;\& \,\lim _{x \to 0} f(x) = 0$

BUT $\displaystyle \lim _{x \to 2} g(f(x)) = 1\;\& \,\lim _{x \to 0} g(f(x))\text{ DNE.}$

Re: Question about finding the limit of a piecewise function of a piecewise function?

Hey all,

This question is free game for anyone. In regards to the diagram referenced in the first post, to which I posted a link, is the value of c that makes $\displaystyle \lim_{x\to1}$[f(x) + c g(x)] exist equal to the same value of c that makes f(x) + c g(x) continuous at x = 1.

Re: Question about finding the limit of a piecewise function of a piecewise function?

I approached the problem this way:

For $\displaystyle \lim_{x\to1}[f(x) + c g(x)]$ to exist, both the left and right limits must exist.

$\displaystyle \lim_{x\to1^-}[f(x)+cg(x)]=4+c(-1)$

$\displaystyle \lim_{x\to1^+}[f(x)+cg(x)]=?$

(The question mark is there not because this limit is unknown, but I leave it to you to figure it out.) Also, the left and right limits must be equal to each other. After finding the second limit, set the two of them equal to each other, and solve for c.

Now refer to the definition of continuity to complete the question, and proceed in a manner similar to above.