Question

decide if the given function is continuous at the specific value of x...

Answer 1

\[f(x)= \frac{ 2x-4 }{ 3x-2 } at x=2\]

Answer 2

remember the definition: f is continous at x=a if
\[ \large \lim_{x\to a}f(x)=f(a) \]

Answer 3

yes it is continuous..

Answer 4

it is defined at x=2 right

Answer 5

f(2)=0
\[\large \lim_{x\to 2}f(x)=0=f(2)\]

Answer 6

and also this is its graph

http://www.wolframalpha.com/input/?i=plot+%282x-4%29%2F%283x-2%29

as you can see the func. is continuous at x=2, no hole no infinity

Answer 7

ok. so if they were ask to list all the value of x for which the given function is not continuous...how would that work?
Example:
\[f(x)= \frac{ 3x-1 }{ 2x-6 }\]

Answer 8

the func. is discontinuous when denominator is 0

Answer 9

2x-6=0
x=3
so, for x=3 the func. is not continuous

Answer 10

and what about the numerator?

Answer 11

it can be any number, no problem with it..

look at its graph

http://www.wolframalpha.com/input/?i=plot+ \frac{+3x-1+}{+2x-6+}

Answer 12

http://www.wolframalpha.com/input/?i=plot+%283x-1%29%2F%282x-6+%29

Answer 13

ok, so there's only one value of x?

Answer 14

in this case yes..

Answer 15

ok. so how about in this case.
f(x) = 3x-2 if x<0
x^2+x if \[x \ge0\]

Answer 16

the func. is not continuous at x=0 why because limit does not exists when we approach to 0, from left limit is -2, from right limit is 0

http://www.wolframalpha.com/input/?i=piecewise+ [{{3x-2%2Cx%3C0}%2C{x^2%2Bx%2Cx%3E%3D0}}]

Answer 17

http://www.wolframalpha.com/input/?i=piecewise [{{3x-2%2Cx%3C0}%2C{x^2%2Bx%2Cx%3E%3D0}}]