Questions: Let the limit as x approaches 7 of f(x) be 18 and the limit as x approaches 7 of g(x) be 15. Use the limit rules to find the limit below. The limit as x approaches 7 of [f(x)-g(x)] What expression results from applying the appropriate limit rule? Find the limit. The limit as x approaches 7 of [f(x)-g(x)] = (Simplify your answer.)

Solution Steps

To find the limit of the expression \(\lim _{x \rightarrow 7}[f(x)-g(x)]\), we can use the limit rule that states the limit of a difference is the difference of the limits. Therefore, we can express the limit as \(\lim _{x \rightarrow 7} f(x) - \lim _{x \rightarrow 7} g(x)\). Given that \(\lim _{x \rightarrow 7} f(x) = 18\) and \(\lim _{x \rightarrow 7} g(x) = 15\), we can substitute these values into the expression to find the limit.

To find the limit of the expression \(\lim _{x \rightarrow 7}[f(x)-g(x)]\), we use the limit rule for the difference of two functions. This rule states that:

\[ \lim _{x \rightarrow a} [f(x) - g(x)] = \lim _{x \rightarrow a} f(x) - \lim _{x \rightarrow a} g(x) \]

We know from the problem that:

\[ \lim _{x \rightarrow 7} f(x) = 18 \quad \text{and} \quad \lim _{x \rightarrow 7} g(x) = 15 \]

Substituting these values into the limit expression gives:

\[ \lim _{x \rightarrow 7}[f(x)-g(x)] = 18 - 15 \]

Now, we perform the subtraction:

\[ 18 - 15 = 3 \]

Final Answer