Questions: Find the following limit or state that it does not exist. lim h → 0 (sqrt(225+h)-15)/h

Find the following limit or state that it does not exist. lim h → 0 (sqrt(225+h)-15)/h
Transcript text: Find the following limit or state that it does not exist. \[ \lim _{h \rightarrow 0} \frac{\sqrt{225+h}-15}{h} \]
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Solution

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Solution Steps

To find the limit, we can use the technique of rationalizing the numerator. Multiply the numerator and the denominator by the conjugate of the numerator, which is \(\sqrt{225+h} + 15\). This will help eliminate the square root in the numerator and simplify the expression, allowing us to evaluate the limit as \(h\) approaches 0.

Step 1: Identify the Limit Expression

We are given the limit expression: \[ \lim _{h \rightarrow 0} \frac{\sqrt{225+h}-15}{h} \]

Step 2: Rationalize the Numerator

To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator, \(\sqrt{225+h} + 15\): \[ \frac{\sqrt{225+h}-15}{h} \cdot \frac{\sqrt{225+h}+15}{\sqrt{225+h}+15} = \frac{(\sqrt{225+h})^2 - 15^2}{h(\sqrt{225+h}+15)} \]

Step 3: Simplify the Expression

Simplify the numerator using the difference of squares: \[ (\sqrt{225+h})^2 - 15^2 = 225 + h - 225 = h \] Thus, the expression becomes: \[ \frac{h}{h(\sqrt{225+h}+15)} = \frac{1}{\sqrt{225+h}+15} \]

Step 4: Evaluate the Limit

Now, evaluate the limit as \(h\) approaches 0: \[ \lim _{h \rightarrow 0} \frac{1}{\sqrt{225+h}+15} = \frac{1}{\sqrt{225}+15} = \frac{1}{15+15} = \frac{1}{30} \]

Final Answer

\(\boxed{\frac{1}{30}}\)

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