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When dealing with limits, the first method you try is often direct substitution. This involves plugging the number that the variable approaches directly into the function. In simple situations, this can quickly give you the limit. To use this method, replace "\( x \)" in the expression with the value it approaches. In the exercise, you substitute \( x = 3 \) into \( \frac{x}{x-3} \). Unfortunately, in this case, direct substitution gives us \( \frac{3}{0} \), which is undefined. That's because division by zero is not allowed in mathematics. It means the expression becomes meaningless, and this results in an undefined form. An undefined result from direct substitution indicates that further analysis is needed, often through other methods.

Limit analysis is crucial when direct substitution leads to an undefined result. This analysis helps determine the behavior of a function as it approaches a specific value. By analyzing the expression \( \frac{x}{x-3} \), it becomes evident that the denominator \( x - 3 \) approaches zero as \( x \) goes towards 3. The numerator still remains 3, making the fraction tend towards infinity because a non-zero number divides by a tiny number.Mathematically, this means the limit doesn't exist in the traditional sense because as you get closer to the point of interest, the values of the function can go towards infinity or negative infinity. Hence, the behavior of \( \frac{x}{x-3} \) indicates that more detailed analysis through one-sided limits is required.

One-sided limits are used when analyzing the behavior of a function as it approaches a certain point from one side—either from above (right) or below (left). These limits can tell us how the function behaves differently from each direction.For \( \lim_{x \rightarrow 3} \frac{x}{x-3} \), if \( x \) approaches 3 from the left (denoted as \( x \rightarrow 3^- \)), the denominator \( x-3 \) takes on small negative values. Thus, the entire expression yields negative infinity, \( -\infty \).Conversely, as \( x \) approaches 3 from the right (denoted as \( x \rightarrow 3^+ \)), \( x-3 \) becomes a small positive number. Hence, the fraction becomes positive, resulting in positive infinity, \( +\infty \).Since these one-sided limits are not equal (\( -\infty \) from the left and \( +\infty \) from the right), the overall limit for these cases does not exist, confirming the conclusion from the initial undefined result.