Questions: Suppose that the limit as t approaches 3 of t times g(t) equals 12. Show that the limit as t approaches 3 of g(t) exists and equals 4.

Solution Steps

To show that \(\lim_{t \rightarrow 3} g(t)\) exists and equals 4, we can use the given limit \(\lim_{t \rightarrow 3} t g(t) = 12\). We can express \(g(t)\) in terms of \(t\) and use the properties of limits to find \(\lim_{t \rightarrow 3} g(t)\).

We are given that: \[ \lim_{t \rightarrow 3} t g(t) = 12 \]

We can express the given limit as: \[ \lim_{t \rightarrow 3} t g(t) = 3 \lim_{t \rightarrow 3} g(t) = 12 \]

To find \( \lim_{t \rightarrow 3} g(t) \), we divide both sides of the equation by 3: \[ 3 \lim_{t \rightarrow 3} g(t) = 12 \implies \lim_{t \rightarrow 3} g(t) = \frac{12}{3} = 4 \]

Final Answer