Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}}$$

Short answer

Answer: The limit of the given expression as \(x \rightarrow 4\) is 0.

Step-by-step solution

Identify the indeterminate form

We want to find the limit of the given expression as \(x \rightarrow 4\). Let's first substitute \(x = 4\) into the expression and see if it is indeterminate: $$\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}} = \frac{3(4-4) \sqrt{4+5}}{3-\sqrt{4+5}} = \frac{0}{3-\sqrt{9}}$$ Since the expression equals \(\frac{0}{0}\), which is an indeterminate form, we must simplify the expression to find the limit as \(x \rightarrow 4\).

Apply rationalizing technique

A common technique to simplify expressions with square roots in the denominator is to rationalize the denominator, which involves multiplying the denominator and the numerator by the conjugate of the denominator. The conjugate of \(3-\sqrt{x+5}\) is \(3+\sqrt{x+5}\). Let's multiply both the numerator and the denominator by the conjugate: $$\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}} \cdot \frac{3+\sqrt{x+5}}{3+\sqrt{x+5}}$$

Simplify the expression

Now let's multiply the numerators and denominators and simplify the expression: $$\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}(3+\sqrt{x+5})}{(3-\sqrt{x+5})(3+\sqrt{x+5})}$$ After multiplying and simplifying, we have: $$\lim _{x \rightarrow 4} \frac{9(x-4)(x+5)}{9-(x+5)}$$ Simplifying further, we get: $$\lim _{x \rightarrow 4} \frac{9(x-4)(x+5)}{4}$$

Evaluate the limit

Now that we have simplified the expression, we can substitute \(x = 4\) into the expression to find the limit: $$\lim _{x \rightarrow 4} \frac{9(4-4)(4+5)}{4} = \frac{9(0)(9)}{4} = 0$$ So, the limit of the given expression as \(x \rightarrow 4\) is 0.

Key concepts

Indeterminate Forms

When evaluating limits, students occasionally encounter expressions that do not immediately reveal the limit as a specific value. These are encapsulated by what mathematicians call indeterminate forms. The classic examples of indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \), \( \infty^0 \), and \( 1^\infty \).

These forms are labeled 'indeterminate' because they do not conclusively signify a specific limit; they actually conceal further behavior that needs to be unveiled through additional steps, such as algebraic manipulation or application of special rules in calculus, like L'Hôpital's Rule. A clear example was seen in the exercise where \( \lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}} = \frac{0}{0} \)—a direct substitution led to an indeterminate form, necessitating further examination to ascertain the actual limit. Simplifying the expression often leads to an accurate determination of the limit.

Rationalizing Denominators

One powerful technique for dealing with limits involving square roots, particularly when we encounter indeterminate forms, is rationalizing the denominator. This process involves multiplying both the numerator and the denominator of the expression by the conjugate of the denominator in order to eliminate the square root from the denominator. The conjugate of a binomial \(a - b\) is \(a + b\), and it is pivotal in this strategy because \( (a - b)(a + b) = a^2 - b^2 \)—a difference of squares which eliminates the square root effectively.

This technique simplifies the limit calculation; in our exercise, we used the conjugate \(3+\sqrt{x+5}\) to transform the original expression, leading to a simplification that ultimately helped us evaluate the limit successfully. Rationalizing the denominator thus helps in converting complex-looking indeterminate forms into simpler algebraic expressions that can be more readily analyzed and simplified.

Simplifying Expressions

The process of simplifying expressions is central to evaluating limits, especially when dealing with indeterminate forms. Simplification may include factoring algebraic expressions, canceling common terms, and expanding products. By breaking down an expression to its simplest form, we can often directly observe the behavior of the function as the variable approaches a specific value.

In the given problem, after rationalizing the denominator, further simplification was needed. The resulting expression was factored and simplified to reveal the limit. Finally, we substituted \(x = 4\) into the simplified expression and found that \(\lim _{x \rightarrow 4} \frac{9(x-4)(x+5)}{4} = 0\). This step was crucial because it eliminated any confusion from the original indeterminate form, \(\frac{0}{0}\), and provided clear evidence of the limit's value. Ensuring each algebraic step is properly performed can help students not only to find the correct limit, but also to comprehend the nature of the function being studied.