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Direct substitution is the simplest method for finding a limit. It involves replacing the variable in a function with the number that the variable is approaching. If the function is well-behaved, or continuous, at that point, you can directly substitute to find the limit.
For example, in the exercise, the expression \( \lim _{x \rightarrow 0} \frac{6x-9}{x^{3}-12x+3} \) was evaluated by substituting 0 for \( x \), resulting in:

• \( 6(0) - 9 \) in the numerator which becomes \(-9\)
• \( 0^3 - 12(0) + 3 \) in the denominator which simplifies to \(3\)

This gave a limit of \( \frac{-9}{3} = -3 \). This substitution step worked because the function was continuous at \( x = 0 \).
Direct substitution is often the first technique used when evaluating limits because it's straightforward and efficient for functions that are continuous at the point in question.

Continuous functions are vital when discussing limits because they allow for direct substitution to be used effectively. A function is continuous at a point if:

• The function is defined at that point.
• The limit of the function as it approaches that point is equal to the value of the function at the point.

Consider the function \( f(x) = \frac{6x-9}{x^{3}-12x+3} \) from the exercise. It was continuous at \( x = 0 \) because:

• Substituting \( x = 0 \) returns a real number, \(-3\).
• The function behaves predictably as it approaches this value, with no sudden breaks or jumps.

If a function is continuous over an interval, direct substitution becomes a reliable method to find limits, simplifying the process significantly.

Evaluating limits is a key part of calculus, providing valuable insight into the behavior of functions as they approach specific points. There are several techniques for evaluating limits, with direct substitution being one of the most common.
In cases where direct substitution results in a determinate form, such as \( \frac{a}{b} \) where \( b eq 0 \), the limit can often be evaluated immediately. In the given exercise:

• Direct substitution led to a non-zero denominator, simplifying the process.
• The function did not require any additional algebraic manipulation.

However, if direct substitution yields an indeterminate form like \( \frac{0}{0} \), other methods such as factorization, rationalization, or L'Hôpital's rule may be necessary to evaluate the limit. Understanding when and how to apply these methods is crucial in calculus.