Question

use lhôpitals rule to find the limit.

$$\lim_{y \to 0} \frac{\sqrt{5y + 16} - 4}{y}$$

$$\lim_{y \to 0} \frac{\sqrt{5y + 16} - 4}{y} = \square$$ (type an integer or a simplified fraction.)

Explanation:

Step1: Check the form of the limit

When \(y = 0\), \(\frac{\sqrt{5y + 16}-4}{y}=\frac{\sqrt{16}-4}{0}=\frac{4 - 4}{0}=\frac{0}{0}\), which is an indeterminate form. So we can apply L'Hôpital's Rule.

Step2: Differentiate the numerator and denominator

The derivative of the numerator \(u=\sqrt{5y + 16}-4=(5y + 16)^{\frac{1}{2}}-4\). Using the chain - rule \(\frac{du}{dy}=\frac{5}{2\sqrt{5y+16}}\).
The derivative of the denominator \(v = y\), \(\frac{dv}{dy}=1\).

Step3: Apply L'Hôpital's Rule

By L'Hôpital's Rule \(\lim_{y
ightarrow0}\frac{\sqrt{5y + 16}-4}{y}=\lim_{y
ightarrow0}\frac{\frac{5}{2\sqrt{5y + 16}}}{1}\).

Step4: Evaluate the limit

Substitute \(y = 0\) into \(\frac{5}{2\sqrt{5y+16}}\), we get \(\frac{5}{2\sqrt{16}}=\frac{5}{8}\).

Answer:

\(\frac{5}{8}\)