Questions: Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. lim (x -> 36) (sqrt(x)-6)/(x-36) A. 6 B. 1/12 C. 1/6 D. 0

Transcript text: Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. \[ \lim _{x \rightarrow 36} \frac{\sqrt{x}-6}{x-36} \] A. 6 B. $\frac{1}{12}$ C. $\frac{1}{6}$ D. 0

Solution

Solution Steps

To solve this limit problem, we can use the technique of rationalizing the numerator. This involves multiplying the numerator and the denominator by the conjugate of the numerator. This will help eliminate the square root and simplify the expression, allowing us to evaluate the limit.

Step 1: Identify the Indeterminate Form

The given limit is $\lim_{x \to 36} \frac{\sqrt{x} - 6}{x - 36}$. Direct substitution of $x = 36$ results in the indeterminate form $\frac{0}{0}$. This indicates that further algebraic manipulation is needed to evaluate the limit.

Step 2: Rationalize the Numerator

To resolve the indeterminate form, multiply the numerator and the denominator by the conjugate of the numerator, $\sqrt{x} + 6$:

\[ \frac{\sqrt{x} - 6}{x - 36} \cdot \frac{\sqrt{x} + 6}{\sqrt{x} + 6} = \frac{(\sqrt{x} - 6)(\sqrt{x} + 6)}{(x - 36)(\sqrt{x} + 6)} \]

Step 3: Simplify the Expression

The numerator simplifies using the difference of squares:

\[ (\sqrt{x} - 6)(\sqrt{x} + 6) = x - 36 \]

Thus, the expression becomes:

\[ \frac{x - 36}{(x - 36)(\sqrt{x} + 6)} = \frac{1}{\sqrt{x} + 6} \]

Step 4: Evaluate the Limit

Now, substitute $x = 36$ into the simplified expression:

\[ \lim_{x \to 36} \frac{1}{\sqrt{x} + 6} = \frac{1}{\sqrt{36} + 6} = \frac{1}{6 + 6} = \frac{1}{12} \]