Question

limit as x approaches 3 of ((x^2)-x-6)/(1-(sqrt(10-3*x)))

Answer 1

I multiplied denominator by 1+sqrt(10-3) to cancel square root and factored out a -3 to get (-3(x+3))

then factoring the top i get (x+2)(x-3) though do not see any cancelation from here due to the fact that I still need to multiply by 1+sqrt(10-3)

Answer 2

after multiplying the denominator you should get \(1-\sqrt{10-3x}^2=1-(10-3x)\)
which is equal to \(3(x-3)\) the (x-3) in this should cancel out with (x-3) on the numerator. Now you should be able to find the limit

Answer 3

which is essentially what I was trying to do, though (possibly i am not seeing it) this does not equal 3(x-3) because 1-10-3x =-9-3x = 3(-x-3)

Answer 4

\[\begin{align*}
(1-\sqrt{10-3x})(1+\sqrt{10-3x})&=1^2-[\sqrt{10-3x}]^2\\
&=1-(10-3x)\\
&=1-10\textbf{+}3x\\
&=3x-9\\
&=3(x-3)
\end{align*}\]

alfterall, you should know that the denominator should have a factor (x-3) because when substituted x=3, you get 0 for the denominator \(1-\sqrt{10-3x}\) (which is the reason to consider algebraic alteration of the expression from the first place)