Find the limit. $$ \lim _{x \rightarrow 144}\left(\frac{x-144}{\sqrt{x}-12}\right) $$

Question

UpStudy AI · Accepted Answer

1. Write the limit:

$$
   \lim_{x \rightarrow 144}\left(\frac{x-144}{\sqrt{x}-12}\right)
   $$

2. Notice that the numerator $$ x-144 $$ can be factored as a difference of squares:

$$
   x-144 = (\sqrt{x}-12)(\sqrt{x}+12)
   $$

3. Substitute this factorization into the expression:

$$
   \frac{x-144}{\sqrt{x}-12} = \frac{(\sqrt{x}-12)(\sqrt{x}+12)}{\sqrt{x}-12}
   $$

4. Cancel the common factor $$ (\sqrt{x}-12) $$ (this cancellation is valid for $$ x \neq 144 $$):

$$
   \frac{(\sqrt{x}-12)(\sqrt{x}+12)}{\sqrt{x}-12} = \sqrt{x}+12
   $$

5. Now, take the limit as $$ x \rightarrow 144 $$:

$$
   \lim_{x \rightarrow 144} (\sqrt{x}+12) = \sqrt{144}+12 = 12+12 = 24
   $$

Therefore, the limit is:

$$
24
$$
The limit is 24.

Find the limit.