Question

Pre-calc, image attached

Answer 1

`\lim_(x \to 7) (√(x+2)-3 )/(x-7) = (1)/(6)`

Explanation:

`\lim_(x \to 7) (√(x+2)-3 )/(x-7)`

`(√(x+2)-3 )/(x-7)`

`\frac{(1)/(2) (x+2)^{-(1)/(2) }}{1}` <-- Take derivative of numerator and denominator expressions

`{(1)/(2) (x+2)^{-(1)/(2) }}` <-- Simplify

`(1)/(2)((1)/(√(x+2) ))` <-- Rewrite

`(1)/(2)((1)/(√(7+2) ))` <-- Use direct substitution and plug in the limit x=7

`(1)/(2)((1)/(√(9) ))`

`(1)/(2)(\frac{1}3 })`

`(1)/(6)`

Therefore,
`\lim_(x \to 7) (√(x+2)-3 )/(x-7) = (1)/(6)`