Limits by double rationalization Calculator

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Example

Created on 2024-06-20Asked by Mateo Thompson (Solvelet student)

Find the limit

\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}

by rationalizing the numerator.

Solution

To find the limit

\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}

by rationalizing the numerator, multiply the numerator and denominator by the conjugate of the numerator:

\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x} \cdot \frac{\sqrt{x + 4} + 2}{\sqrt{x + 4} + 2}

This gives:

\lim_{x \to 0} \frac{(\sqrt{x + 4} - 2)(\sqrt{x + 4} + 2)}{x(\sqrt{x + 4} + 2)}

The numerator simplifies to:

(\sqrt{x + 4})^2 - 2^2 = x + 4 - 4 = x

So the limit becomes:

\lim_{x \to 0} \frac{x}{x(\sqrt{x + 4} + 2)} = \lim_{x \to 0} \frac{1}{\sqrt{x + 4} + 2}

Now, substitute

x = 0

\lim_{x \to 0} \frac{1}{\sqrt{0 + 4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}

Therefore, the limit is:

\frac{1}{4}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Aria Adams on Solvelet

1. Evaluate

\lim_{x \to 1} \frac{\sqrt{2x + 3} - \sqrt{5x - 2}}{x - 1}

by double rationalization,2. Find

\lim_{x \to 0} \frac{x^2 - 4}{x^2 + 2x}

by factoring.,

DefinitionLimits by Double Rationalization CalculatorDefinition: A calculator to solve limits by double rationalizing using its conjugate to simplify the radical expression. Here is an example of how to solve \( \lim_{{x \to 0}} \frac{\sqrt{x+4} - 2}{x} \) : \[ = \frac{(x+4) - 4}{x(\sqrt{x+4} + 2)} = \frac{x}{x(\sqrt{x+4} + 2)} = \frac{1}{\sqrt{x+4} + 2} \] As \(x \to 0\), \((\frac{1}{\sqrt{4}+2} = \frac{1}{4}\))

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