Question

Limits involving conjugates Evaluate the following limits. $$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3}$$

Short answer

$$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4x+5}-3}$$ Answer: The limit of the given function as x approaches 1 is \(\frac{3}{2}\).

Step-by-step solution

Identify the conjugate of the expression containing the radical

We have \(\sqrt{4x+5}-3\) in the denominator. The conjugate of this expression is \(\sqrt{4x+5}+3\).

Multiply the numerator and denominator by the conjugate

Multiply both the numerator and denominator of the fraction by the conjugate, \(\sqrt{4x+5}+3\): $$\frac{x-1}{\sqrt{4x+5}-3} \times \frac{\sqrt{4x+5}+3}{\sqrt{4x+5}+3}$$

Simplify the expression

Perform the multiplication and simplify the expression: $$\frac{(x-1)(\sqrt{4x+5}+3)}{(\sqrt{4x+5}-3)(\sqrt{4x+5}+3)} = \frac{(x-1)(\sqrt{4x+5}+3)}{(4x+5)-9} = \frac{(x-1)(\sqrt{4x+5}+3)}{4x-4}$$ Now, factor out a 4 from the denominator to get: $$\frac{(x-1)(\sqrt{4x+5}+3)}{4(x-1)}$$

Cancel out common factors

Cancel out the common factors \((x-1)\) from the numerator and denominator: $$\frac{\cancel{(x-1)}(\sqrt{4x+5}+3)}{4\cancel{(x-1)}} = \frac{\sqrt{4x+5}+3}{4}$$

Evaluate the limit

Now, we can evaluate the limit as x approaches 1: $$\lim _{x \rightarrow 1} \frac{\sqrt{4x+5}+3}{4} = \frac{\sqrt{4(1)+5}+3}{4} = \frac{\sqrt{9}+3}{4} = \frac{6}{4} = \frac{3}{2}$$ Thus, the limit of the given function as x approaches 1 is \(\frac{3}{2}\).

Key concepts

Conjugate Method

In calculus, the conjugate method is a useful strategy for simplifying expressions, especially those involving radicals or square roots. When evaluating limits, such expressions can sometimes pose challenges due to their complexity or indeterminate forms, such as \(\frac{0}{0}\). To quickly tackle these, we use their conjugates.

The conjugate of a number is formed by switching the sign between two terms. In our example, the conjugate of \(\sqrt{4x+5}-3\) is \(\sqrt{4x+5}+3\). When you multiply an expression by its conjugate, the result is typically a simpler or more manageable expression, often removing the radical.

This technique helps us to eliminate discrepancies or undefined behaviors by converting difficult denominator expressions into a polynomial form. After this transformation, we can easily simplify and evaluate limits without division by zero.

Rationalizing Techniques

Rationalizing techniques are essential in handling expressions involving radicals. By rationalizing the numerator or denominator, we convert irrational expressions into rational ones, smoothing the path for further calculations.

In rationalizing, we often multiply by the conjugate. Consider the original problem \(\frac{x-1}{\sqrt{4x+5}-3}\). We multiply by the conjugate \(\frac{\sqrt{4x+5}+3}{\sqrt{4x+5}+3}\). This action results in a difference of squares, simplifying our denominator to \(4x-4\), which is a straightforward expression.

This step is quintessential to not just simplify, but also to ensure that the output is a rational expression free from radicals. Once the radicals have been rationalized, further simplification steps become obvious and seamless.

Limits with Radicals

Understanding limits with radicals is an important aspect of calculus. These problems often involve indeterminate forms, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), requiring us to find creative ways to simplify.

With radicals, direct substitution often fails as it yields undefined or indeterminate forms. Here, the conjugate method becomes incredibly useful. By applying the method systematically, we change the structure of our limit expression to exclude undefined terms.

After rationalizing and eliminating common factors, as seen in \(\lim_{x \rightarrow 1} \frac{\sqrt{4x+5}+3}{4}\), the expression becomes much neater. We can then apply direct substitution to finalize the limit value. This technique ensures we confidently handle limits involving otherwise tricky radicals, leading to a clear answer without errors.