Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3}$$

Short answer

Answer: The limit of the function is \(\frac{3}{2}\) as x approaches 1.

Step-by-step solution

Identify the indeterminate form

Verify the given form is \(\frac{0}{0}\) when plugging in x = 1: $$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3} = \frac{1-1}{\sqrt{4 \cdot 1+5}-3} = \frac{0}{0}$$ Since the form is indeterminate, we must find another method to compute the limit.

Rationalize the denominator

Rationalize the denominator, by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{4x+5}+3\). $$\lim _{x \rightarrow 1} \frac{(x-1)(\sqrt{4 x+5}+3)}{(\sqrt{4 x+5}-3)(\sqrt{4x+5}+3)}$$

Simplify the expression

Now, we'll simplify the expression by multiplying out the numerator and denominator: $$\lim _{x \rightarrow 1} \frac{(x-1)(\sqrt{4 x+5}+3)}{(4x + 5) - 3^2}$$ $$\lim _{x \rightarrow 1} \frac{(x-1)(\sqrt{4 x+5}+3)}{4x - 4}$$

Factor the denominator

Factor the denominator as \(4(x - 1)\) to cancel out the common factors in the numerator and the denominator: $$\lim _{x \rightarrow 1} \frac{(x-1)(\sqrt{4 x+5}+3)}{4(x - 1)}$$ $$\lim _{x \rightarrow 1} \frac{\cancel{(x-1)}(\sqrt{4 x+5}+3)}{4\cancel{(x - 1)}}$$ $$\lim _{x \rightarrow 1} \frac{\sqrt{4 x+5}+3}{4}$$

Find the limit

Now the expression can be evaluated directly by substituting x = 1: $$\lim _{x \rightarrow 1} \frac{\sqrt{4 x+5}+3}{4} = \frac{\sqrt{4(1)+5}+3}{4} = \frac{\sqrt{9}+3}{4} = \frac{6}{4}$$

Final Answer

$$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3} = \frac{3}{2}$$

Key concepts

Indeterminate Forms

When attempting to find the limit of a function as a variable approaches a certain value, we may encounter indeterminate forms like \(\frac{0}{0}\). These forms occur when substituting the value into both the numerator and denominator results in zero. For example, in the given exercise, as \(x\) approaches 1, both the numerator \((x - 1)\) and the denominator \((\sqrt{4x+5} - 3)\) become zero. This produces an indeterminate form \(\frac{0}{0}\), indicating more work is needed to find the limit.

Indeterminate forms can be quite challenging as they do not provide any information about the actual limit value. Thus, we need to simplify the expression using techniques such as rationalization or substitution.

Rationalization

Rationalization is a method often used to simplify expressions involving radicals. In the given problem, the denominator has a square root expression \((\sqrt{4x+5} - 3)\). By multiplying the numerator and denominator by the conjugate \((\sqrt{4x+5} + 3)\), we can eliminate the radicals.

Multiplying by the conjugate becomes:

• Numerator: \((x-1)(\sqrt{4x+5} + 3)\)
• Denominator: \((4x+5) - 3^2\)

This results in a simpler expression, allowing us to factor and cancel terms. By eliminating the radical, we make it possible to directly evaluate the limit.

Substitution

Substitution is a useful strategy once the expression is simplified and terms have been canceled. In our example, after rationalizing, we end up with a simplified form where the \((x - 1)\) term cancels out. The resulting expression \(\frac{\sqrt{4x+5} + 3}{4}\) can now be evaluated by directly substituting \(x = 1\).

Substitution helps to finalize the process by calculating the limit. After all the simplifications, we find:

• \(\lim _{x \rightarrow 1} \frac{\sqrt{4(1)+5} + 3}{4} = \frac{\sqrt{9} + 3}{4} = \frac{6}{4}\)

This simple substitution reveals the final limit value of \(\frac{3}{2}\).

Conjugates

Conjugates are pairs of expressions like \((a - b)\) and \((a + b)\), which are useful for eliminating radicals. In our exercise, the denominator has \(\sqrt{4x+5} - 3\), and its conjugate is \(\sqrt{4x+5} + 3\). Multiplying an expression by its conjugate helps remove the radical by employing the difference of squares identity: \((a-b)(a+b) = a^2 - b^2\).

In this step-by-step solution, we multiplied the expression by the conjugate to change the form of the denominator:

• The resulting expression was \((4x+5) - 9\) or \(4x - 4\), a polynomial term.

Using conjugates simplifies the calculation by making it easier to identify common factors or substitutions. This makes evaluating the limit straightforward, revealing the desired limit value efficiently.