`lim_(x to oo) (sqrt(x + 1) - sqrt(x))` equals

Options

A. `oo`
B. 0
C. `-1`
D. 1

Answer

To solve the limit \( \lim_{x \to \infty} (\sqrt{x + 1} - \sqrt{x}) \), we will follow these steps:

### Step 1: Identify the form of the limit
As \( x \) approaches infinity, both \( \sqrt{x + 1} \) and \( \sqrt{x} \) approach infinity. Thus, we have an indeterminate form of type \( \infty - \infty \).

**Hint:** When you encounter an indeterminate form, consider algebraic manipulation such as rationalization.

### Step 2: Rationalize the expression
To resolve the indeterminate form, we can multiply and divide by the conjugate of the expression:
\[ \lim_{x \to \infty} \left( \sqrt{x + 1} - \sqrt{x} \right) \cdot \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} \]

This gives us:
\[ \lim_{x \to \infty} \frac{(\sqrt{x + 1} - \sqrt{x})(\sqrt{x + 1} + \sqrt{x})}{\sqrt{x + 1} + \sqrt{x}} = \lim_{x \to \infty} \frac{(x + 1) - x}{\sqrt{x + 1} + \sqrt{x}} \]

**Hint:** The numerator simplifies to a difference of squares.

### Step 3: Simplify the numerator
The numerator simplifies to:
\[ 1 \]
Thus, we have:
\[ \lim_{x \to \infty} \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

**Hint:** Focus on the dominant terms in the limit as \( x \) approaches infinity.

### Step 4: Analyze the denominator
As \( x \) approaches infinity, both \( \sqrt{x + 1} \) and \( \sqrt{x} \) approach \( \sqrt{x} \). Therefore, the denominator simplifies to:
\[ \sqrt{x + 1} + \sqrt{x} \approx \sqrt{x} + \sqrt{x} = 2\sqrt{x} \]

**Hint:** When \( x \) is very large, \( \sqrt{x + 1} \) behaves similarly to \( \sqrt{x} \).

### Step 5: Substitute back into the limit
Now we can rewrite the limit:
\[ \lim_{x \to \infty} \frac{1}{\sqrt{x + 1} + \sqrt{x}} = \lim_{x \to \infty} \frac{1}{2\sqrt{x}} \]

**Hint:** As \( x \) approaches infinity, consider how the expression behaves.

### Step 6: Evaluate the limit
As \( x \) approaches infinity, \( \sqrt{x} \) approaches infinity, so:
\[ \frac{1}{2\sqrt{x}} \to 0 \]

Thus, we conclude:
\[ \lim_{x \to \infty} (\sqrt{x + 1} - \sqrt{x}) = 0 \]

### Final Answer:
The limit is \( 0 \).

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