Limits by rationalizing Calculator

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Example

Created on 2024-06-20Asked by Lucas White (Solvelet student)

Find the limit

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

by rationalizing the numerator.

Solution

To find the limit

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

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

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

This gives:

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

The numerator simplifies to:

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

So the limit becomes:

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

Now, substitute

x = 4

\lim_{x \to 4} \frac{1}{\sqrt{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 Layla Thompson on Solvelet

1. Find the limit

\lim_{{x \to 0}} \frac{{1 - \cos(x)}}{{x^2}}

using rationalizing.2. Evaluate the limit

\lim_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x}

using rationalizing.

DefinitionHow to convert limits to a rationalizing form where you are basically dealing with square roots by multiplying the numerator and denominator by the conjugate of the numerator or denominator. To solve for lim_{x->0}frac{x}x+1-frac1x+1, we will need to multiply by frac{x+1}{x+1} and then further simplify the limit.

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