Questions: The initial substitution of x=a yields the form 0/0. Simplify the function algebraically, or use a table or graph to determine the limit. If necessary, state that the limit does not exist. lim as x approaches 18 of (x^2+3x-378)/(x^2-324) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim as x approaches 18 of (x^2+3x-378)/(x^2-324)= (Type an integer or a simplified fraction.)

Solution Steps

To solve the limit problem, we first need to simplify the expression algebraically. The given expression is a rational function, and the initial substitution of \( x = 18 \) results in an indeterminate form \( \frac{0}{0} \). To resolve this, we can factor both the numerator and the denominator and then cancel out any common factors. After simplification, we can substitute \( x = 18 \) to find the limit.

The given limit is

\[ \lim _{x \rightarrow 18} \frac{x^{2}+3x-378}{x^{2}-324} \]

Substituting \( x = 18 \) directly into the expression results in the indeterminate form \( \frac{0}{0} \).

To resolve the indeterminate form, we factor both the numerator and the denominator:

• The numerator \( x^2 + 3x - 378 \) factors to \( (x - 18)(x + 21) \).
• The denominator \( x^2 - 324 \) factors to \( (x - 18)(x + 18) \).

After factoring, the expression becomes:

\[ \frac{(x - 18)(x + 21)}{(x - 18)(x + 18)} \]

We can cancel the common factor \( (x - 18) \) from the numerator and the denominator:

\[ \frac{x + 21}{x + 18} \]

Now, we substitute \( x = 18 \) into the simplified expression:

\[ \frac{18 + 21}{18 + 18} = \frac{39}{36} = \frac{13}{12} \]

Final Answer