To solve this math problem, we will substitute t = 3 into the given function M(t) = 25 + 85e^(-0.48t) to find the number of words still remembered after 3 days.

(1) The number of words still remembered after 3 days is:
M(3) = 25 + 85e^(-0.48*3)
M(3) = 25 + 85e^(-1.44)
M(3) ≈ 25 + 85 * 0.2369
M(3) ≈ 25 + 20.1385
M(3) ≈ 45.1385

Therefore, the number of words still remembered after 3 days is approximately 45.1385.

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To find M'(3), we need to take the derivative of M(t) with respect to t and substitute t = 3 into the resulting derivative function. M'(t) represents the rate of change of the number of words remembered with respect to time.

M(t) = 25 + 85e^(-0.48t)

Taking the derivative of M(t) with respect to t using the chain rule:
M'(t) = (-0.48 * 85) * e^(-0.48t)

Substituting t = 3:
M'(3) = (-0.48 * 85) * e^(-0.48*3)
M'(3) ≈ -40.8 * 0.2369
M'(3) ≈ -9.67872

Therefore, M'(3) is approximately -9.67872.

The negative value of M'(3) indicates that the rate of forgetting the French vocabulary words is approximately 9.67872 words per day after 3 days of studying. This means that, on average, the person forgets around 9.67872 words each day after 3 days of studying.

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To find the limit of M(t) as t approaches infinity, we need to evaluate the function as t approaches positive infinity.

The formula for M(t) is:
M(t) = 25 + 85e^(-0.48t)

As t approaches infinity, the exponential term e^(-0.48t) approaches 0. Therefore, the limit of M(t) as t approaches infinity is:

lim t→∞ M(t) = 25 + 85 * 0
lim t→∞ M(t) = 25

This means that as time goes on indefinitely (approaching infinity), the number of words still remembered approaches a constant value of 25.

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