Questions: Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim t→0 (5/t - 5/(t^2 + t))

Transcript text: Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) \[ \lim _{t \rightarrow 0}\left(\frac{5}{t}-\frac{5}{t^{2}+t}\right) \]

Solution

Solution Steps

To evaluate the limit, we first need to simplify the expression by combining the fractions into a single fraction. Then we can substitute $t = 0$ into the simplified expression to find the limit.

Step 1: Simplify the expression

Combine the fractions to get a single fraction: \[ \frac{5}{t} - \frac{5}{t^2 + t} = \frac{5(t^2 + t) - 5t}{t(t^2 + t)} = \frac{5t^2 + 5t - 5t}{t(t^2 + t)} = \frac{5t^2}{t(t^2 + t)} = \frac{5t}{t^2 + 1} \]

Step 2: Substitute $ t = 0 $ into the simplified expression

Substitute $ t = 0 $ into the simplified expression: \[ \lim_{t \to 0} \frac{5t}{t^2 + 1} = \frac{5 \times 0}{0^2 + 1} = 0 \]