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

Solution:

1. Substitution to check if the limit can be directly evaluated: First, substitute $x = 1$ into the expression:

   $\frac{\sqrt{1} - \sqrt{1 + 3}}{1 - (1)^2} = \frac{1 - 2}{1 - 1} = \frac{-1}{0}$ This gives an indeterminate form $\frac{-1}{0}$, which suggests that direct substitution does not work. Therefore, we need to simplify the expression using algebraic methods.

2. Factoring the denominator: Notice that the denominator can be factored as: $1 - x^2 = (1 - x)(1 + x)$

3. Rationalizing the numerator: To simplify the numerator, we multiply the expression by the conjugate of the numerator: $\frac{\sqrt{x} - \sqrt{x + 3}}{(1 - x)(1 + x)} \cdot \frac{\sqrt{x} + \sqrt{x + 3}}{\sqrt{x} + \sqrt{x + 3}}$ This simplifies the numerator using the identity $(a - b)(a + b) = a^2 - b^2$: $\frac{(\sqrt{x})^2 - (\sqrt{x + 3})^2}{(1 - x)(1 + x)(\sqrt{x} + \sqrt{x + 3})} = \frac{x - (x + 3)}{(1 - x)(1 + x)(\sqrt{x} + \sqrt{x + 3})}$ Simplify the numerator: $\frac{x - x - 3}{(1 - x)(1 + x)(\sqrt{x} + \sqrt{x + 3})} = \frac{-3}{(1 - x)(1 + x)(\sqrt{x} + \sqrt{x + 3})}$

4. Canceling terms: Notice that $1 - x$ appears in both the numerator and denominator, so we can cancel it (changing the sign): $\frac{3}{(1 + x)(\sqrt{x} + \sqrt{x + 3})}$

5. Substitute $x = 1$: Now, substitute $x = 1$ into the simplified expression: $\frac{3}{(1 + 1)(\sqrt{1} + \sqrt{1 + 3})} = \frac{3}{(2)(1 + 2)} = \frac{3}{6} = \frac{1}{2}$

Final Answer:

$\lim_{{x \to 1}} \frac{\sqrt{x} - \sqrt{x + 3}}{1 - x^2} = \frac{1}{2}$

Would you like a deeper explanation of any part of the solution, or do you have any questions?

Related Questions:

1. What is the significance of rationalizing the numerator in limits?
2. How can we approach limits that result in indeterminate forms?
3. How does L'Hopital's Rule apply to limits of indeterminate forms like $\frac{0}{0}$?
4. What happens if we approach the limit from the left-hand or right-hand side?
5. Can this method of rationalizing be applied to other types of functions involving square roots?

Tip:

Always check if the limit can be simplified algebraically before applying more complex methods like L'Hopital's Rule.