f(x) = (x-2)/(x+3)
show that x is continuous at x=1

Question

f(x) = (x-2)/(x+3)
show that x is continuous at x=1

anonymous · Answer

Do I need to show that lim x->1 (f(x)) == f(1) ?

anonymous · Answer

yes.
lim[x->1] f(x)=f(1)
and that lim[x->1] f(x) exists, saying that both sides are equal.
and 3rd condition, that f(1) exists, but once you get the first 1, the last 2 are obvious.

anonymous · Answer

$$\lim_{x \rightarrow 1}\frac{x-2}{x+3}=f(1)$$

anonymous · Answer

$$\lim_{x \rightarrow 1}\frac{(x)-2}{(x)+3}=\frac{(1)-2}{(1)+3}=\frac{-1}{4}=-1/4$$

anonymous · Answer

so, you can ell right away that lim{x->1} f(x) is equal to f(1),
since there are no restrictions for x to equal 1.

it is continuous, fo all numbers besides x=-3, right?
(vertical asymptote.)

anonymous · Answer

Sounds right, the prof wants us to show by comparing the limit to f(x) I think.  How would you prove if there were a jump in the graph?  Would you have to show that it is different from the left and right, as well as at the point itself?