Question

Find the limit, if it exists (picture below)

Answer 1

c.
`(1)/(2 √(7) )`

Explanation:

When plugging in zero into the given equation:

`\lim_(x \rightarrow 0) (√(x + 7) - √(7) )/(x) = (0)/(0)`

Answer is in indeterminate form = use L'Hospital's Rule:

(Derivative of the top / Derivative of the bottom)

`\lim_(x \rightarrow 0) \frac{ (1)/(2) (x + 7)^{ (-1)/(2)} - 0 }{1}`

Rearranged equation:

`\lim_(x \rightarrow 0) (1)/(2 √(x + 7) )`

Plug zero back into equation:

`\lim_(x \rightarrow 0) (1)/(2 √(x + 7) ) = (1)/(2 √(0 + 7) ) = (1)/(2 √(7) )`

Answer:

`\lim_(x \rightarrow 0) (√(x + 7) - √(7) )/(x) = (1)/(2 √(7) )`