Question

Please help me please

Answer 1

`-(1)/(64)`

Explanation:

Evaluate the following limit.

`\lim_(x \to 0) ((1)/(x+8) -(1)/(8) )/(x)`

(1) - Simplify the limit

`\lim_(x \to 0) ((1)/(x+8) -(1)/(8) )/(x)\\\\\Longrightarrow \lim_(x \to 0) ((1(8))/((x+8)(8)) -(1(x+8))/(8(x+8)) )/(x)\\\\\Longrightarrow \lim_(x \to 0) ((8-x-8)/(8(x+8)) )/(x) \\\\\Longrightarrow \lim_(x \to 0) (( -x)/(8(x+8)) )/(x) \\\\\Longrightarrow \lim_(x \to 0) (-x)/(8x(x+8)) \\\\\Longrightarrow \boxed{\lim_(x \to 0) (-1)/(8(x+8)) }`

(2) - Plug in the limit

`\lim_(x \to 0) (-1)/(8(x+8))\\\\\Longrightarrow \lim_(x \to 0) (-1)/(8((0)+8))\\\\\Longrightarrow \lim_(x \to 0) (-1)/(8(8)) \\\\\therefore \boxed{\boxed{\lim_(x \to 0) ((1)/(x+8) -(1)/(8) )/(x)=-(1)/(64) }}`