1. ## Limits

In the following library of functions, what are the values of $\displaystyle c$ for which $\displaystyle \lim_{x\to{c}}f(x)=f(c)$?

Polynomial function:

$\displaystyle f(x)=a_nx^n+...+a_1x+a_0$

Rational function:

$\displaystyle f(x)=\frac{p(x)}{q(x)}$

Trigonomic functions:

$\displaystyle f(x)=\sin{x}$, $\displaystyle f(x)\cos{x}$,
$\displaystyle f(x)=\tan{x}$, $\displaystyle f(x)=\cot{x}$,
$\displaystyle f(x)=\sec{x}$ ,$\displaystyle f(x)=\csc{x}$,

Exponential functions:

$\displaystyle f(x)=a^x$ $\displaystyle f(x)=e^x$

Natural logarithmic functions:

$\displaystyle f(x)=lnx$

I just kind of threw this out there guys because I'd like to improve my understanding of limits and continuity. Anyone who wants to tackle these has a thanks coming. More than one person's views are always welcome, so if anyone sees that the questions already been answered, don't let that stop you putting your spin on it. Thanks guys.

2. Originally Posted by VonNemo19
In the following library of functions, what are the values of $\displaystyle c$ for which $\displaystyle \lim_{x\to{c}}f(x)=f(c)$?

Polynomial function:

$\displaystyle f(x)=a_nx^n+...+a_1x+a_0$

Rational function:

$\displaystyle f(x)=\frac{p(x)}{q(x)}$

Trigonomic functions:

$\displaystyle f(x)=\sin{x}$, $\displaystyle f(x)\cos{x}$,
$\displaystyle f(x)=\tan{x}$, $\displaystyle f(x)=\cot{x}$,
$\displaystyle f(x)=\sec{x}$ ,$\displaystyle f(x)=\csc{x}$,

Exponential functions:

$\displaystyle f(x)=a^x$ $\displaystyle f(x)=e^x$

Natural logarithmic functions:

$\displaystyle f(x)=lnx$

I just kind of threw this out there guys because I'd like to improve my understanding of limits and continuity. Anyone who wants to tackle these has a thanks coming. More than one person's views are always welcome, so if anyone sees that the questions already been answered, don't let that stop you putting your spin on it. Thanks guys.
It will be true whenever $\displaystyle f(x)$ is continous at c becuse that is the defintion of continuity at a point! i.e

A function $\displaystyle f(x)$ is continous at $\displaystyle c$ if
$\displaystyle f(c)=\lim_{x \to c}f(x)$

so for the first one it is always true because polynomials are continous on the whole real line.

2. It will be true anytime $\displaystyle g(c) \ne =0$

....

I won't do the rest but what I said first will carry over to all of the other examples.

3. What's the definiton of continuity over (a,b) then?

4. Originally Posted by VonNemo19
What's the definiton of continuity over (a,b) then?
If it is continous at each $\displaystyle x \in (a,b)$

5. So, if I hear you correctly, if $\displaystyle \lim_{x\to{c}}f(x)=f(c)$ for any and every value of c in (a,b), then $\displaystyle f(x)$ is said to be continuous over (a,b). Can this be tested or only implied?

6. Originally Posted by VonNemo19
So, if I hear you correctly, if $\displaystyle \lim_{x\to{c}}f(x)=f(c)$ for any and every value of c in (a,b), then $\displaystyle f(x)$ is said to be continuous over (a,b). Can this be tested or only implied?
yes that is correct. Since any non empty inveteral that is not degenerate i.e $\displaystyle [a,a]$ (only has one point) is uncountable there is no way to "test every point"

So we have a bunch of functions we "know" are continous i.e like polynomials(and others on you list), and the fact that the product, sum, difference and quotient (as long as the denominator is not 0) of continous function is continous.

7. Thanks Man.

What made you decide on the new photo?It's cool.