91Ó°ÊÓ

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 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 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 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.