Evaluate the following limit: lim (x->4) (√(x) - 2) / (√(x^3) - 8)
Understand the Problem
The question asks us to evaluate the limit of a function as x approaches 4. The function involves square roots in both the numerator and the denominator. The limit is of the form 0/0, which means we may need to rationalize or factor to simplify the expression.
Answer
$\frac{1}{12}$
Answer for screen readers
$\frac{1}{12}$
Steps to Solve
- Recognize indeterminate form
First, substitute $x = 4$ into the expression to check if it yields an indeterminate form.
$\frac{\sqrt{4} - 2}{\sqrt{4^3} - 8} = \frac{2 - 2}{\sqrt{64} - 8} = \frac{0}{8 - 8} = \frac{0}{0}$
Since we have the indeterminate form $\frac{0}{0}$, we can proceed to simplify the expression.
- Rewrite the expression
Rewrite $\sqrt{x^3}$ as $(\sqrt{x})^3$ and $8$ as $2^3$:
$\lim_{x \to 4} \frac{\sqrt{x} - 2}{(\sqrt{x})^3 - 2^3}$
- Factor the denominator
Use the difference of cubes factorization formula, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where $a = \sqrt{x}$ and $b = 2$.
$(\sqrt{x})^3 - 2^3 = (\sqrt{x} - 2)((\sqrt{x})^2 + 2\sqrt{x} + 2^2) = (\sqrt{x} - 2)(x + 2\sqrt{x} + 4)$
Now, rewrite the limit:
$\lim_{x \to 4} \frac{\sqrt{x} - 2}{(\sqrt{x} - 2)(x + 2\sqrt{x} + 4)}$
- Simplify the expression
Cancel the common factor $(\sqrt{x} - 2)$ from the numerator and the denominator:
$\lim_{x \to 4} \frac{1}{x + 2\sqrt{x} + 4}$
- Evaluate the limit
Substitute $x = 4$ into the simplified expression:
$\frac{1}{4 + 2\sqrt{4} + 4} = \frac{1}{4 + 2(2) + 4} = \frac{1}{4 + 4 + 4} = \frac{1}{12}$
$\frac{1}{12}$
More Information
The limit of the function as $x$ approaches $4$ is $\frac{1}{12}$.
Tips
A common mistake is incorrectly factoring the denominator or not recognizing the difference of cubes pattern. Another mistake could be not checking for the indeterminate form before attempting to simplify the expression.
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