The limit of `[(1)/(x)sqrt(1+x)-sqrt(1+(1)/(x^(2))]]` as ` x rarr0`

Options

A. does not exist
B. is equal to 1/2
C. is equal to 0
D. is equal to 1

Answer

To find the limit of the expression

\[ \lim_{x \to 0} \left( \frac{1}{x \sqrt{1+x}} - \sqrt{1+\frac{1}{x^2}} \right), \]

we will simplify the expression step by step.

### Step 1: Rewrite the expression
We start by rewriting the limit:

\[ L = \lim_{x \to 0} \left( \frac{1}{x \sqrt{1+x}} - \sqrt{1+\frac{1}{x^2}} \right). \]

### Step 2: Find a common denominator
To combine the two fractions, we need a common denominator. The common denominator will be \( x \sqrt{1+x} \):

\[ L = \lim_{x \to 0} \left( \frac{1 - x \sqrt{1+\frac{1}{x^2}} \cdot \sqrt{1+x}}{x \sqrt{1+x}} \right). \]

### Step 3: Simplify the square root term
Next, we simplify the square root term:

\[ \sqrt{1+\frac{1}{x^2}} = \sqrt{\frac{x^2 + 1}{x^2}} = \frac{\sqrt{x^2 + 1}}{x}. \]

Substituting this back into our limit gives:

\[ L = \lim_{x \to 0} \left( \frac{1 - \sqrt{x^2 + 1} \cdot \sqrt{1+x}}{x \sqrt{1+x}} \right). \]

### Step 4: Expand the square root using Taylor series
As \( x \to 0 \), we can use the Taylor series expansion for \( \sqrt{1+x} \) and \( \sqrt{x^2+1} \):

\[ \sqrt{1+x} \approx 1 + \frac{x}{2}, \quad \sqrt{x^2 + 1} \approx 1 + \frac{x^2}{2}. \]

Substituting these approximations into the limit gives:

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

### Step 5: Simplify the numerator
Now simplify the numerator:

\[ 1 - \left( 1 + \frac{x^2}{2} + \frac{x}{2} + \frac{x^3}{4} \right) = -\frac{x^2}{2} - \frac{x}{2} - \frac{x^3}{4}. \]

### Step 6: Substitute back into the limit
Now substitute this back into the limit:

\[ L = \lim_{x \to 0} \left( \frac{-\frac{x^2}{2} - \frac{x}{2} - \frac{x^3}{4}}{x(1 + \frac{x}{2})} \right). \]

### Step 7: Factor out \( x \)
Factoring out \( x \) from the numerator gives:

\[ L = \lim_{x \to 0} \left( \frac{-\frac{x}{2} - \frac{1}{2} - \frac{x^2}{4}}{1 + \frac{x}{2}} \right). \]

### Step 8: Evaluate the limit
As \( x \to 0 \):

\[ L = \frac{-\frac{0}{2} - \frac{1}{2} - 0}{1 + 0} = -\frac{1}{2}. \]

Thus, the limit is:

\[ \boxed{-\frac{1}{2}}. \]