anonymous
  • anonymous
Check if the function is continuous.
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
\[f(x) = \left\{ {{{x^2 + 6x - 16} \over {x^2 + x -6}}, x \neq 2} \right\}\] \[{7x - 4}, x =2\]
anonymous
  • anonymous
I can't order the brackets
myininaya
  • myininaya
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

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myininaya
  • myininaya
oh x=2 is defined for 7x-4
myininaya
  • myininaya
still since the limit doesn't exist at x=2 then f is not continuous there
anonymous
  • anonymous
yeap. sorry for that.
anonymous
  • anonymous
Oh! OK! Thanks.
myininaya
  • myininaya
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
nikvist
  • nikvist
\[\lim_{x\rightarrow 2}f(x)=\lim_{x\rightarrow 2}\frac{x^2+6x-16}{x^2+x-6}=\lim_{x\rightarrow 2}\frac{(x+8)(x-2)}{(x+3)(x-2)}=\lim_{x\rightarrow 2}\frac{x+8}{x+3}=2\] \[f(2)=7\cdot 2-4=10\] \[\lim_{x\rightarrow 2}f(x)\neq f(2)\Rightarrow f(x)\quad not\enspace continuous\]
anonymous
  • anonymous
Thanks!

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