Question

Limits involving conjugates Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{x}{\sqrt{c x+1}-1}, \text { where } c \text { is a constant }$$

Short answer

Question: Evaluate the limit of the expression \(\frac{x}{\sqrt{cx+1}-1}\) as \(x \rightarrow 0\), where c is a constant. Answer: The limit as \(x \rightarrow 0\) is \(\frac{2}{c}\).

Step-by-step solution

Multiply by the conjugate of the denominator

To rationalize the denominator, we'll multiply both the numerator and the denominator by the conjugate of the denominator, i.e., \(\sqrt{cx+1}+1\): $$\frac{x}{\sqrt{cx+1}-1} \cdot \frac{\sqrt{cx+1}+1}{\sqrt{cx+1}+1}$$

Simplify the expression

Now, we'll multiply the expressions and simplify the numerator and the denominator. First, let's focus on the denominator: $$(\sqrt{cx+1}-1)(\sqrt{cx+1}+1) = (\sqrt{cx+1})^2 - 1^2 = cx+1 - 1 = cx$$ Our expression now becomes: $$\frac{x(\sqrt{cx+1}+1)}{cx}$$

Cancel out common factors

We notice that there is a common factor of x in both the numerator and the denominator. Let's cancel this out: $$\frac{\cancel{x}(\sqrt{cx+1}+1)}{c\cancel{x}} = \frac{\sqrt{cx+1}+1}{c}$$

Evaluate the limit

Now, we can easily evaluate the limit as \(x \rightarrow 0\): $$\lim_{x \rightarrow 0} \frac{\sqrt{cx+1}+1}{c} = \frac{\sqrt{c \cdot 0 + 1} + 1}{c} = \frac{\sqrt{1} + 1}{c} = \frac{2}{c}$$ Hence, the limit as \(x \rightarrow 0\) is \(\frac{2}{c}\), with c being a constant.

Key concepts

Conjugates

In mathematics, a conjugate is a term that resembles another, but with an opposite sign between two parts. For example, if you have \(a + b\), its conjugate would be \(a - b\). Conjugates are particularly useful for simplifying expressions, especially when dealing with square roots or complex numbers.

In this specific problem, we use the conjugate to deal with the square root in the denominator. By multiplying the expression \(\frac{x}{\sqrt{cx+1}-1}\) by the conjugate of the denominator, which is \(\sqrt{cx+1}+1\), we transform the problem into a simpler form.

• When multiplying conjugates, the result is the difference of squares: \(a^2 - b^2\).
• This helps to eliminate the square root and simplify the fraction.

Using conjugates is a clever algebraic trick to simplify certain types of limits and integrals, making them easier to evaluate.

Rationalization

Rationalization is a technique that is often applied to remove irrational numbers (like square roots) from the denominator of a fraction. This process makes the expression easier to handle and more straightforward to evaluate.

In the exercise, the fraction \(\frac{x}{\sqrt{cx+1}-1}\) contains a square root in the denominator, complicating direct evaluation. By multiplying by the conjugate \(\sqrt{cx+1}+1\), rationalization is achieved:

• This process turns the irrational denominator into a rational expression, in this case, \(cx\).
• With a rationalized denominator, it's straightforward to perform further algebraic simplifications.

After rationalization, the expression becomes \(\frac{x(\sqrt{cx+1}+1)}{cx}\), which is much easier to simplify, allowing us to confidently cancel out terms and evaluate the limit.

Calculus

Calculus is the mathematical study of continuous change and includes two main branches: differentiation and integration. In the context of this problem, we're dealing with limits, a fundamental concept in calculus.

Evaluating limits often requires precise algebraic manipulation, such as using conjugates and rationalization. Here's a concise breakdown of how these concepts are utilized:

• Limits: Understanding limits involves approaching a value without necessarily reaching it. For this exercise, we explore how \(x\) gets infinitely close to 0 without actually being 0.
• Avoiding Direct Substitution: Directly substituting might initially lead to an indeterminate form, which necessitates further manipulation through techniques like conjugation and rationalization.

The essence of calculus here lies in identifying and applying appropriate methods to simplify complex expressions, facilitating accurate evaluations of limits to determine the exact behavior of functions as inputs approach specified points.