Questions: Let f(x)=(x^2-4x-21)/(x=7). Find a) limx -> 7 f(x) and b) limx -> 0 f(x). a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. limx -> 7 f(x)=□ (Simplify your answer.) B. The limit does not exist. b) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. limx -> 0 f(x)=□ (Simplify your answer.) B. The limit does not exist.

Let f(x)=(x^2-4x-21)/(x=7). Find a) limx -> 7 f(x) and b) limx -> 0 f(x).
a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. limx -> 7 f(x)=□ (Simplify your answer.)
B. The limit does not exist.
b) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. limx -> 0 f(x)=□ (Simplify your answer.)
B. The limit does not exist.

Transcript text: Let $f(x)=\frac{x^{2}-4 x-21}{x=7}$. Find a) $\lim _{x \rightarrow 7} f(x)$ and b) $\lim _{x \rightarrow 0} f(x)$. a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow 7} f(x)=\square$ (Simplify your answer.) B. The limit does not exist. b) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow 0} f(x)=\square$ (Simplify your answer.) B. The limit does not exist.

Solution

Solution Steps

Step 1: Evaluate $\lim_{x \rightarrow 7} f(x)$

To find $\lim_{x \rightarrow 7} f(x)$, we first simplify the function $ f(x) = \frac{x^2 - 4x - 21}{x - 7} $. We factor the numerator:

\[ x^2 - 4x - 21 = (x - 7)(x + 3) \]

Thus, we can rewrite $ f(x) $ as:

\[ f(x) = \frac{(x - 7)(x + 3)}{x - 7} \]

For $ x \neq 7 $, this simplifies to:

\[ f(x) = x + 3 \]

Now, we can evaluate the limit:

\[ \lim_{x \rightarrow 7} f(x) = \lim_{x \rightarrow 7} (x + 3) = 7 + 3 = 10 \]

Step 2: Evaluate $\lim_{x \rightarrow 0} f(x)$

Next, we find $\lim_{x \rightarrow 0} f(x)$. Since we have already simplified $ f(x) $ to $ x + 3 $ for $ x \neq 7 $, we can directly substitute $ x = 0 $:

\[ \lim_{x \rightarrow 0} f(x) = 0 + 3 = 3 \]