# Continuity

• Mar 14th 2013, 03:51 PM
raggie29
Continuity
Please, please please help me with this questions. Really struggling as i have been off uni ill for over a week now and missed alot of classes to make it worse this assesment is due in asap!! :(

Many thanks, Hanna. x
• Mar 14th 2013, 04:15 PM
HallsofIvy
Re: Continuity
It will certainly be impossible to do this without knowing the definition of "continuous at x= a"! And that is that $\displaystyle \lim_{x\to a} f(x)= f(a)$. You will also need to know that the limit exists only if the limits, from above and below, both exist and are equal.

So for the first problem, to determine whether or not $\displaystyle f(x)= -2x^3+ 3x^2$ if $\displaystyle x\le 1$, $\displaystyle f(x)= sin((\pi/2)x)$ if x> 1, You need to show that
$\displaystyle \lim_{x\to 1} -2x^3+ 3x^2$ and $\displaystyle \lim_{x\to 1} sin((\pi/2)x)$ are the same and equal to f(1).

(For the last one you need to think about $\displaystyle (x- 1)^2 sin\left(\frac{1}{x- 1}\right)= (x- 1)\frac{sin\left(\frac{1}{x-1}\right)}{\frac{1}{x-1}}$.
What do you remember about $\displaystyle lim_{a\to 0}\frac{sin(a)}{a}$?