Questions: lim as x approaches 0 of x / (sqrt(x+3) - sqrt(3))

Transcript text: \(\lim _{x \rightarrow 0} \frac{x}{\sqrt{x+3}-\sqrt{3}}\)

Solution

Solution Steps

Step 1: Rationalization

To rationalize the expression, multiply and divide by the conjugate of the numerator: \(\frac{\sqrt{x+3} + sqrt(3)}{\sqrt{x+3} + sqrt(3)}\). This gives us: \[\frac{\left(\sqrt{x+3} - sqrt(3)\right)\left(\sqrt{x+3} + sqrt(3)\right)}{x\left(\sqrt{x+3} + sqrt(3)\right)} = \frac{\sqrt{x + 3} - \sqrt{3}}{x}\]

Step 2: Direct Substitution

After simplification, we directly substitute \(x = 0\) into the simplified expression to find the limit. \[\lim_{x \to 0}\frac{\sqrt{x + 3} - \sqrt{3}}{x} = 0.289\]

Final Answer:

The limit as \(x\) approaches 0 is 0.289.

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