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L'Hopital's Rule is a powerful technique in calculus used to find limits of indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When direct substitution in a limit results in these forms, L'Hopital's Rule allows us to differentiate the numerator and denominator separately and then compute the limit again.

In the given exercise, the limit \(\lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16}\) poses an indeterminate form of \(\frac{0}{0}\) when \(x = 4\). Here, both the numerator and denominator approach zero. By applying L'Hopital's Rule, we differentiate the numerator, \(3(x-4)\), which gives us \(3\), and the denominator, \(x^2 - 16\), giving \(2x\).

Now, substitute these derivatives back into the limit to get \(\lim _{x \rightarrow 4} \frac{3}{2x}\). Substitute \(x = 4\) to find the limit, which simplifies to \(\frac{3}{8}\).

L'Hopital's Rule is especially helpful when functions become complex or direct substitution fails.

The factoring technique is a simple algebraic method used to simplify expressions before evaluating limits. It involves rewriting a polynomial expression as a product of its factors. This technique can help cancel terms that are causing an indeterminate form, making it easier to find the limit.

In the problem \(\lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16}\), the denominator \(x^2 - 16\) can be factored using the difference of squares formula, \((a^2 - b^2 = (a+b)(a-b))\). Here it becomes \((x+4)(x-4)\).

The original expression becomes \(\frac{3(x-4)}{(x+4)(x-4)}\). You can then cancel \(x-4\) from the numerator and denominator to get \(\frac{3}{x+4}\). Now, the indeterminate portion is removed, allowing for simple substitution of \(x = 4\) to yield \(\lim _{x \rightarrow 4} \frac{3}{x+4} = \frac{3}{8}\).

Factoring is particularly useful for simplifying rational expressions and solving limits without advanced calculus techniques.

An indeterminate form is a mathematical expression that initially seems to be undefined, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms arise naturally when computing limits because both parts of the fraction behave similarly while approaching a particular point.

In the given limit problem, substituting \(x = 4\) into \(\lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16}\) yields \(\frac{0}{0}\), a classic indeterminate form. Such forms necessitate additional strategies, like L'Hopital's Rule or algebraic manipulation, to resolve.

Recognizing indeterminate forms is crucial because it guides the selection of techniques to turn the expression into a determinate form. Once transformed, straightforward calculations can be applied to determine the limit.

This is significant because indeterminate forms can conceal underlying behavior of functions, which these techniques help to reveal.

Calculus techniques provide comprehensive tools to analyze and solve limits, derivatives, and integrals. Beyond manual computation, these techniques involve a logical set of steps to address specific mathematical challenges.

In the realm of limits, two key techniques are factoring and using derivatives as seen with L'Hopital's Rule. These techniques can either simplify expressions to a workable form or reduce complex limits to simple arithmetic.

The use of these techniques as demonstrated in the exercise, showcases their effectiveness:

• Factoring eliminated an indeterminate form through algebraic simplification.
• L'Hopital's Rule addressed a \(\frac{0}{0}\) indeterminate by differentiation.

In calculus, diverse situations demand different approaches, and having a robust understanding of these techniques is invaluable for resolving complex mathematical problems effectively.