91Ó°ÊÓ

Evaluating limits is a foundational concept in calculus. It allows us to understand the behavior of a function as it approaches a particular point. A limit is expressed as \( \lim_{x \to a} f(x) \), where the goal is to determine what value \( f(x) \) gets closer to as \( x \) approaches \( a \). This concept is crucial when a function isn't defined at a certain point but might still approach a specific value there. To correctly evaluate a limit, analyze the function at and around the given point. Here are some things to consider:

• Assess whether directly substituting the value of \( x \) gives a valid result.
• Alternatively, check for an indeterminate form, which might require further methods to evaluate properly.

In the example exercise, evaluating the limit \( \lim _{x \rightarrow 0} \frac{6x - 9}{x^{3} - 12x + 3} \), we successfully substituted \( x = 0 \) into the function and found a valid limit without complications.

Indeterminate forms occur when substituting the given value into a function results in expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( 0^0 \), among others. These forms imply that the behavior of the function is ambiguous at the point of calculation, and require careful analysis or additional methods to resolve.
To tackle indeterminate forms:

• You may need to factorize, rationalize, or apply L'Hôpital's Rule.
• Look for algebraic simplifications that resolve the indeterminate nature.

In the given problem, when substituting \( x = 0 \) in \( \frac{6x - 9}{x^{3} - 12x + 3} \), the result was \( \frac{-9}{3} \), which is \(-3\). This indicates no indeterminate form appeared, simplifying the evaluation process.

The substitution method is a straightforward technique for evaluating limits where a direct approach can be used. When evaluating a limit, if substituting the value of \( x \) does not lead to an indeterminate form, you can directly compute the limit. This method is efficient and time-saving when applicable.
Steps for using substitution effectively:

• Substitute the value directly into the function, if possible.
• After inputting the value, simplify the expression to find the limit.

In our problem, substituting \( x = 0 \) into the expression \( \frac{6(0) - 9}{0^3 - 12(0) + 3} \), the calculation simplifies to \(-3\). This result shows how the substitution method can quickly provide an answer when no complications arise.