Question

Check if the function is continuous.

Answer 1

\[f(x) = \left\{ {{{x^2 + 6x - 16} \over {x^2 + x -6}}, x \neq 2} \right\}\]
\[{7x - 4}, x =2\]

Answer 2

I can't order the brackets

Answer 3

x^2+x-6=(x+3)(x-2) x^2+6x-16 doesn't have either factor so the limit does not exsit at -3 and 2 so the function is not continuous at either x

Answer 4

oh x=2 is defined for 7x-4

Answer 5

still since the limit doesn't exist at x=2 then f is not continuous there

Answer 6

yeap. sorry for that.

Answer 7

Oh! OK! Thanks.

Answer 8

all you have to do is make sure the limit exists and the limx->af(x)=f(a) then f is continuous at x=a

Answer 9

\[\lim_{x\rightarrow 2}f(x)=\lim_{x\rightarrow 2}\frac{x^2+6x-16}{x^2+x-6}=\lim_{x\rightarrow 2}\frac{(x+8)(x-2)}{(x+3)(x-2)}=\lim_{x\rightarrow 2}\frac{x+8}{x+3}=2\]
\[f(2)=7\cdot 2-4=10\]
\[\lim_{x\rightarrow 2}f(x)\neq f(2)\Rightarrow f(x)\quad not\enspace continuous\]

Answer 10

Thanks!