Question

Evaluate the following limits. \(\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}}\)

Short answer

Answer: The limit of the given function as x approaches 4 is -54.

Step-by-step solution

Identify the function and the limiting value of x

First, let's identify the function and the limiting value of x: Function: \(f(x) = \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}}\) Limiting value of x: \(x \rightarrow 4\)

Check if direct substitution works

Try to substitute the limiting value of x into the function to see if it results in indeterminate form: \(f(4) = \frac{3(4-4) \sqrt{4+5}}{3-\sqrt{4+5}} = \frac{0}{0}\) As the result is an indeterminate form, we need to manipulate the function algebraically so that we can evaluate the limit.

Manipulate the function algebraically

Recall that the conjugate of \(a - b\) is \(a + b\). To remove the square root from the denominator, multiply both the numerator and denominator by the conjugate of the denominator (\(3 + \sqrt{x+5}\)): \(f(x) = \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}} \cdot \frac{3+\sqrt{x+5}}{3+\sqrt{x+5}} = \frac{3(x-4) \sqrt{x+5}(3 + \sqrt{x+5})}{(3-\sqrt{x+5})(3+\sqrt{x+5})}\)

Simplify the denominator

Use the difference of squares formula to simplify the denominator: \((3-\sqrt{x+5})(3+\sqrt{x+5}) = 3^2 - (\sqrt{x+5})^2 = 9 - (x+5) = 4-x\)

Factor out (x-4) from the numerator

Factor out \((x-4)\) from the numerator: \(f(x) = \frac{3(x-4) \left[\sqrt{x+5}(3 + \sqrt{x+5})\right]}{4-x}\)

Rewrite and simplify the function

Rewrite every instance of x as \(4+(x-4)\) and factor out a negative sign from the denominator to make it look like the numerator: \(f(x) = \frac{3(x-4) \left[\sqrt{4+1+(x-4)}(3 + \sqrt{4+1+(x-4)})\right]}{-(x-4)} = -3 \left[\sqrt{9+(x-4)}(3 + \sqrt{9+(x-4)})\right]\)

Substitute the limiting value of x

Now, we can substitute the limiting value of x into the function: \(f(4) = -3 \left[\sqrt{9+(4-4)}(3 + \sqrt{9+(4-4)})\right] = -3 \left[\sqrt{9}(3 + \sqrt{9})\right]\)

Calculate the limit

Finally, compute the value of the function as x approaches 4: \(\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}} = -3\left[\sqrt{9}(3 + \sqrt{9})\right] = -3(3)(6)=-54\) The limit of the given function as x approaches 4 is -54.

Key concepts

Conjugate Multiplication

When evaluating limits involving square roots, one of the effective techniques is conjugate multiplication. This technique helps us remove the square root by multiplying the numerator and the denominator by the conjugate of the denominator. Conjugates come in pairs, such as \(a-b\) and \(a+b\). This method is beneficial because multiplying these pairs results in the difference of squares, a pattern that eliminates square roots.
The process becomes simpler when aiming to simplify the expression, enabling you to break down complex functions in stages.

• Identify the conjugate of the term with the square root.
• Multiply both numerator and denominator by this conjugate.
• Simplify using the difference of squares rule, leading to a more simplified form ready for limit evaluation.

Conjugate multiplication transforms complex expressions, making the journey toward solving limits more manageable. It's a powerful tool when handling indeterminate forms, especially ones involving irrational terms.

Indeterminate Forms

Indeterminate forms are expressions which involve limits that initially appear undefined. These often appear in forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Such expressions are common roadblocks in calculus, but they can often be resolved using techniques like algebraic manipulation or L'Hôpital's Rule.
In the problem we're dealing with, substituting the limiting value directly yields an indeterminate form \(\frac{0}{0}\). This tells us that the limit cannot be directly evaluated simply by plugging in the value.

• Identify if a direct substitution leads to an indeterminate form.
• Use algebraic techniques such as function simplification or application of conjugates to resolve the indeterminacy.
• Re-evaluate the simplified expression to obtain a defined limit.

Understanding how to manage indeterminate forms is crucial, as it enables students to handle seemingly undefined functions and extract meaningful results through thoughtful manipulation.

Difference of Squares

The difference of squares is a valuable algebraic pattern used for simplifying expressions, often enabling the removal of square roots from denominators. This pattern is expressed as \(a^2 - b^2 = (a-b)(a+b)\), showing that the product of a sum and difference of the same numbers results in a subtraction of their squares.
This principle is commonly employed in rationalizing denominators. In our original exercise, applying the difference of squares formula helps remove the square root, simplifying the denominator significantly.

• Recognize expressions fitting the difference of squares pattern.
• Apply the formula to eliminate roots and simplify complex fractions.
• Use the simplified form to re-evaluate limits that were previously indeterminate.

By mastering the difference of squares, students can streamline otherwise complicated algebraic expressions, paving the way for clearer evaluations of limits and other functions.

Function Simplification

Simplifying a function is often necessary when direct substitution results in an undefined or complicated expression. In calculus, this allows for a clearer analysis of the function's behavior, particularly when evaluating limits.
The goal is to transform the function into an equivalent, but more manageable form, facilitating the limit evaluation process. Function simplification can include factoring, canceling terms, or leveraging algebraic identities.

• Identify parts of the function that can be factored or simplified.
• Use algebraic rules, such as factoring and identity principles, to simplify.
• After simplification, re-evaluate the function to determine the limit.

Effective function simplification not only resolves indeterminate forms but also provides insight into the function's behavior. It's a vital skill in students' mathematical toolkit, empowering them to tackle complex problems with confidence.