Evaluate the limit as t approaches 5 for the expression (t - sqrt(4t + 5))/(5 - t).

To evaluate the limit as ( t ) approaches 5 for the expression ( \frac{t - \sqrt{4t + 5}}{5 - t} ), we first substitute ( t = 5 ) directly into the expression. This gives us:

[

\frac{5 - \sqrt{4(5) + 5}}{5 - 5} = \frac{5 - \sqrt{20 + 5}}{0} = \frac{5 - \sqrt{25}}{0} = \frac{5 - 5}{0} = \frac{0}{0}.

]

The result is an indeterminate form, which means we need to simplify the expression to evaluate the limit.

To simplify, we can multiply the numerator and denominator by the conjugate of the numerator. The conjugate of ( t - \sqrt{4t + 5} ) is ( t + \sqrt{4t + 5} ). Therefore, we rewrite the limit as:

[

\lim_{t \to 5} \frac{(t - \sqrt{4t + 5})(t + \sqrt{4t + 5})}{