Evaluate `lim_(xto0) {(sinx-x+(x^(3))/(6))/(x^(5))}.`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^5}, \]

we can follow these steps:

### Step 1: Identify the form of the limit
When we substitute \(x = 0\) into the expression, we get:

\[ \sin(0) - 0 + \frac{0^3}{6} = 0 - 0 + 0 = 0, \]

and the denominator \(x^5\) also approaches \(0\). Thus, we have a \( \frac{0}{0} \) indeterminate form.

**Hint:** Check the value of the numerator and denominator as \(x\) approaches 0 to confirm the indeterminate form.

### Step 2: Use the Taylor series expansion for \(\sin x\)
The Taylor series expansion for \(\sin x\) around \(x = 0\) is:

\[ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \cdots \]

Substituting this into our limit gives:

\[ \sin x - x + \frac{x^3}{6} = \left(x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots\right) - x + \frac{x^3}{6}. \]

**Hint:** Remember the Taylor series expansion for \(\sin x\) to simplify the expression.

### Step 3: Simplify the expression
The terms \(x\) and \(-x\) cancel out, and we are left with:

\[ \sin x - x + \frac{x^3}{6} = \frac{x^5}{120} - \frac{x^7}{5040} + \cdots \]

Thus, we can rewrite our limit as:

\[ \lim_{x \to 0} \frac{\frac{x^5}{120} - \frac{x^7}{5040} + \cdots}{x^5}. \]

**Hint:** Combine like terms and factor out \(x^5\) from the numerator.

### Step 4: Factor out \(x^5\)
Now we can factor out \(x^5\) from the numerator:

\[ \lim_{x \to 0} \left( \frac{1}{120} - \frac{x^2}{5040} + \cdots \right). \]

**Hint:** After factoring, observe how the limit behaves as \(x\) approaches 0.

### Step 5: Evaluate the limit
As \(x\) approaches \(0\), all terms involving \(x\) will vanish, leaving us with:

\[ \frac{1}{120}. \]

Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^5} = \frac{1}{120}. \]

**Final Answer:**

\[ \frac{1}{120}. \]