Question

Use the information to evaluate the limits. \(\lim _{x \rightarrow c} f(x)=2\) \(\lim _{x \rightarrow c} g(x)=3\) (a) \(\lim _{x \rightarrow c}[5 g(x)]\) (b) \(\lim _{x \rightarrow c}[f(x)+g(x)]\) (c) \(\lim _{x \rightarrow c}[f(x) g(x)]\) (d) \(\lim _{x \rightarrow c} \frac{f(x)}{g(x)}\)

Short answer

The limits are: (a) 15, (b) 5, (c) 6, and (d) \(\frac{2}{3}\).

Step-by-step solution

Find the Limit of a Constant times a Function

We start with the first part \((a) \lim _{x \rightarrow c}[5 g(x)]\). Using the Limit of a Constant times a Function property, we multiply the constant 5 by the limit of \(g(x)\) as \(x\) approaches \(c\): \(5 * \lim _{x \rightarrow c} g(x) = 5 * 3 = 15\).

Find the Limit of a Sum

Next, we look at part \((b) \lim _{x \rightarrow c}[f(x)+g(x)]\). Using the Limit of a Sum property, we add the limits of \(f(x)\) and \(g(x)\) as \(x\) approaches \(c\): \(\lim _{x \rightarrow c} f(x) + \lim _{x \rightarrow c} g(x) = 2 + 3 = 5\).

Find the Limit of a Product

Part \((c) \lim _{x \rightarrow c}[f(x) g(x)]\) requires the Limit of a Product property. We multiply the limits of \(f(x)\) and \(g(x)\) as \(x\) approaches \(c\): \(\lim _{x \rightarrow c} f(x) * \lim _{x \rightarrow c} g(x) = 2 * 3 = 6\).

Find the Limit of a Quotient

Finally, in part \((d) \lim _{x \rightarrow c} \frac{f(x)}{g(x)}\), we apply the Limit of a Quotient property. We divide the limit of \(f(x)\) by the limit of \(g(x)\) as \(x\) approaches \(c\), given that \(\lim _{x \rightarrow c} g(x)\) is not zero: \(\lim _{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim _{x \rightarrow c} f(x)}{\lim _{x \rightarrow c} g(x)} = \frac{2}{3}\).