Questions: Consider the function h(t) = (9 cos(t) - 9) / t^2. Can you plug t=0 into this function? No Yes Fill in the table below to look at the behavior near t=0. Enter UNDEFINED if the function is undefined. Round answers to three decimal places. t h(t) 1 0.5 0.1 0.01 0 -0.01 -0.1 -0.5 -1 Estimate the value of lim t -> 0 h(t) = If the limit does not exist, write DNE. Round your answer to one decimal place.

Solution Steps

To analyze the behavior of the function \( h(t) = \frac{9 \cos(t) - 9}{t^2} \) near \( t = 0 \), we need to evaluate the function at several points close to zero. Since the function is undefined at \( t = 0 \), we will calculate \( h(t) \) for values approaching zero from both the positive and negative sides. This will help us estimate the limit as \( t \) approaches zero. We will round the results to three decimal places and determine the limit to one decimal place.

We are given the function \( h(t) = \frac{9 \cos(t) - 9}{t^2} \) and need to evaluate it at several points near \( t = 0 \). The function is undefined at \( t = 0 \) because it results in division by zero. We calculate \( h(t) \) for the following values of \( t \):

• \( t = 1 \), \( h(1) = -4.137 \)
• \( t = 0.5 \), \( h(0.5) = -4.407 \)
• \( t = 0.1 \), \( h(0.1) = -4.496 \)
• \( t = 0.01 \), \( h(0.01) = -4.500 \)
• \( t = 0 \), \( h(0) = \text{UNDEFINED} \)
• \( t = -0.01 \), \( h(-0.01) = -4.500 \)
• \( t = -0.1 \), \( h(-0.1) = -4.496 \)
• \( t = -0.5 \), \( h(-0.5) = -4.407 \)
• \( t = -1 \), \( h(-1) = -4.137 \)

To estimate the limit of \( h(t) \) as \( t \) approaches 0, we observe the values of \( h(t) \) for \( t \) close to zero. The values of \( h(t) \) for \( t = 0.01 \) and \( t = -0.01 \) are both approximately \(-4.500\).

The behavior of \( h(t) \) as \( t \) approaches zero from both the positive and negative sides suggests that the limit is approximately \(-4.500\). Therefore, we estimate:

\[ \lim_{t \to 0} h(t) = -4.5 \]

Final Answer