To find the limit of the expression
[tex]\[
\lim _{b \rightarrow 4} \frac{2 b}{\sqrt{5 b + 5} - 1},
\][/tex]
we first observe that it has a form that can potentially lead us to an indeterminate expression. To simplify the process, we should look for an algebraic manipulation that helps us to work with this limit more effectively.
Let's start by simplifying the expression inside the limit. If we plug in [tex]\( b = 4 \)[/tex] directly, we get:
[tex]\[
\frac{2 \times 4}{\sqrt{5 \times 4 + 5} - 1} = \frac{8}{\sqrt{20 + 5} - 1} = \frac{8}{\sqrt{25} - 1} = \frac{8}{5 - 1} = \frac{8}{4} = 2.
\][/tex]
Therefore, the limit as [tex]\( b \)[/tex] approaches 4 of the given function is:
[tex]\[
\lim _{b \rightarrow 4} \frac{2 b}{\sqrt{5 b + 5} - 1} = 2.
\][/tex]
Thus, the final answer is:
[tex]\[
2.
\][/tex]
