lim t>7 F(x) (1/t)−(1/7)/t-7
To evaluate the limit as t approaches 7 of the function F(x) = (1/t) - (1/7) / (t - 7), we can follow these steps:
Step 1: Simplify the expression inside F(x).
F(x) = (1/t) - (1/7) / (t - 7)
= (1/t) - (1/7) / (t - 7) * (t/7)
= (1/t) - (t/7) / (t - 7)
Step 2: Combine the fractions inside F(x) over a common denominator.
F(x) = (7 - t) / (7t(t - 7))
Step 3: Substitute t = 7 into the expression.
F(7) = (7 - 7) / (7 * 7 * (7 - 7))
= 0 / (0 * 0)
= 0 / 0
Notice that we end up with the indeterminate form of 0/0.
Step 4: Apply L'Hopital's Rule, if applicable.
To use L'Hopital's Rule, we differentiate the numerator and the denominator separately with respect to t.
Differentiating the numerator:
d(7 - t) / dt = -1
Differentiating the denominator:
d(7t(t - 7)) / dt = 7(t - 7) + 7t = 14t - 49
Step 5: Replace the original expression with the result of the derivatives.
F(x) = -1 / (14t - 49)
Step 6: Evaluate the limit as t approaches 7.
lim t→7 F(x) = lim t→7 -1 / (14t - 49)
Substituting t = 7:
lim t→7 F(x) = -1 / (14(7) - 49)
= -1 / (98 - 49)
= -1 / 49
Therefore, the limit as t approaches 7 of the function F(x) is -1/49.