Evaluate the limit: \[ \lim _{t \rightarrow 0} \frac{\sin (8 t)}{10 t} \] Upload Choose a File

Step 1 ：We are given the limit to evaluate: \(\lim _{t \rightarrow 0} \frac{\sin (8 t)}{10 t}\)

Step 2 ：We notice that as t approaches 0, the function is in the form of 0/0, which is an indeterminate form.

Step 3 ：We can apply L'Hopital's rule in this case. L'Hopital's rule states that the limit of a function in the form of 0/0 or ∞/∞ can be found by taking the derivative of the numerator and the derivative of the denominator and then finding the limit of the ratio of these derivatives.

Step 4 ：First, we find the derivative of the numerator, \(\sin(8t)\), which is \(8\cos(8t)\).

Step 5 ：Next, we find the derivative of the denominator, \(10t\), which is 10.

Step 6 ：We then find the limit of the ratio of these derivatives as t approaches 0, which gives us \(\lim _{t \rightarrow 0} \frac{8\cos(8t)}{10}\).

Step 7 ：As t approaches 0, \(\cos(8t)\) approaches 1, so the limit is \(\frac{8}{10} = 0.8\).

Step 8 ：Final Answer: The limit of the function as t approaches 0 is \(\boxed{0.8}\)