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

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.

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.

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

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

The numerator simplifies using the difference of squares:

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

Thus, the expression becomes:

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

Now, substitute \(x = 36\) into the simplified expression:

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

Final Answer