Questions: The function (f) has a removable discontinuity at (c). Define (f(c)) so that (f) is continuous at (c). (f(x)=fracx^2+3 x-18x-3, quad c=3) (Give an exact answer. Use symbolic notation and fractions where needed.) (f(c)=)

The function (f) has a removable discontinuity at (c). Define (f(c)) so that (f) is continuous at (c). (f(x)=fracx^2+3 x-18x-3, quad c=3) (Give an exact answer. Use symbolic notation and fractions where needed.) (f(c)=)
Transcript text: The function $f$ has a removable discontinuity at $c$. Define $f(c)$ so that $f$ is continuous at $c$. \[ f(x)=\frac{x^{2}+3 x-18}{x-3}, \quad c=3 \] (Give an exact answer. Use symbolic notation and fractions where needed.) \[ f(c)= \] $\square$
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Solution

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Solution Steps

Step 1: Factor the Numerator

The given function is

\[ f(x) = \frac{x^2 + 3x - 18}{x - 3} \]

To remove the discontinuity at \(x = 3\), we first factor the numerator:

\[ x^2 + 3x - 18 = (x - 3)(x + 6) \]

Step 2: Simplify the Function

By canceling the common factor \((x - 3)\) in the numerator and the denominator, the function simplifies to:

\[ f(x) = x + 6 \]

Step 3: Evaluate the Limit as \(x\) Approaches 3

To define \(f(c)\) so that \(f\) is continuous at \(c = 3\), we evaluate the limit of the simplified function as \(x\) approaches 3:

\[ \lim_{x \to 3} (x + 6) = 3 + 6 = 9 \]

Final Answer

\(\boxed{f(c) = 9}\)

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