Question

((1/x-4)-(1/3))/(x-7) find the limit as x approaches 7

Answer 1

Answer: -1/9

Explanation:

`\displaystyle\\\lim\limits_(x \to 7) ((1)/(x-4) -(1)/(3) )/(x-7) \\ Uncertainty \ (0)/(0) \\\\Let's\ simplify\ the\ sublimit\ expression:\\\\((1)/(x-4) -(1)/(3) )/(x-7)\\\\Simplify\ the\ numerator:\\\\(1)/(x-4)-(1)/(3)=\\\\ (1*3-1*(x-4))/(3(x-4)) =\\\\(3-x+4)/(3(x-4)) =\\\\(7-x)/(3(x-4)) =\\\\(-(x-7))/(3(x-4)) \\\\Hence,\\\\((-(x-7))/(3(x-4)) )/(x-7) =\\-(1)/(3(x-4)) \\`

`Thus,\\\displaystyle\\ \lim\limits_(x \to 7) (-(1)/(3(x-4)))=\\\\-(1)/(3(7-4)) =\\\\-(1)/(3*3) =\\\\-(1)/(9)`