Questions: Find the limit. lim as h approaches 0 of (sqrt(19h+1)-1)/h

Solution Steps

To find the limit of the given expression as \( h \) approaches 0, we can use the technique of rationalizing the numerator. This involves multiplying the numerator and the denominator by the conjugate of the numerator. This will help eliminate the square root and simplify the expression, allowing us to evaluate the limit.

We are given the limit expression: \[ \lim _{h \rightarrow 0} \frac{\sqrt{19h+1}-1}{h} \]

To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator, \(\sqrt{19h+1} + 1\): \[ \frac{\sqrt{19h+1}-1}{h} \cdot \frac{\sqrt{19h+1}+1}{\sqrt{19h+1}+1} = \frac{(19h+1) - 1}{h(\sqrt{19h+1}+1)} \]

Simplify the numerator: \[ 19h \] Thus, the expression becomes: \[ \frac{19h}{h(\sqrt{19h+1}+1)} \]

Cancel \( h \) from the numerator and the denominator: \[ \frac{19}{\sqrt{19h+1}+1} \] Now, evaluate the limit as \( h \) approaches 0: \[ \lim _{h \rightarrow 0} \frac{19}{\sqrt{19h+1}+1} = \frac{19}{\sqrt{1}+1} = \frac{19}{2} \]

Final Answer