Question

1. find the limit.

lim_{x \to 0} \frac{x^{2}-x+sin x}{2x}

Explanation:

Step1: Split the fraction

$\lim_{x
ightarrow0}\frac{x^{2}-x + \sin x}{2x}=\lim_{x
ightarrow0}(\frac{x^{2}}{2x}-\frac{x}{2x}+\frac{\sin x}{2x})$

Step2: Simplify each term

$\lim_{x
ightarrow0}(\frac{x^{2}}{2x}-\frac{x}{2x}+\frac{\sin x}{2x})=\lim_{x
ightarrow0}(\frac{x}{2}-\frac{1}{2}+\frac{\sin x}{2x})$

Step3: Use limit properties

$\lim_{x
ightarrow0}(\frac{x}{2}-\frac{1}{2}+\frac{\sin x}{2x})=\lim_{x
ightarrow0}\frac{x}{2}-\lim_{x
ightarrow0}\frac{1}{2}+\frac{1}{2}\lim_{x
ightarrow0}\frac{\sin x}{x}$

Step4: Evaluate each limit

We know that $\lim_{x
ightarrow0}\frac{x}{2}=0$, $\lim_{x
ightarrow0}\frac{1}{2}=\frac{1}{2}$ and $\lim_{x
ightarrow0}\frac{\sin x}{x} = 1$.
So, $0-\frac{1}{2}+\frac{1}{2}\times1$

Answer:

$0$