Question

Use the properties of limits and the facts given below to find each of the limits given Solve the last one

Answer 1

Given:

`\lim_(x\to0)\text{ }\frac{sin^2x\text{ + sinx\lparen cosx -1\rparen}}{x^2}`

Apply L' Hospital's rule:

`=\lim_(x\to0)\text{ }\frac{sin(2x)\text{ + cosx\lparen cosx - 1\rparen-sin}^2x}{2x}`

Apply L'Hospital's rule:

`=\lim_(x\to0)\text{ }\frac{2cos(2x)\text{ -2sin\lparen2x\rparen+ sin\lparen x\rparen}}{2}`

Plug the value of 0 for x:

`\begin{gathered} =\text{ }\frac{2cos0\text{ - 2sin0 + sin0}}{2} \\ =\text{ }(2)/(2) \\ =\text{ 1} \end{gathered}`

Answer: 1