Determine the following limits.
lim x→3^- x − 4 / x^2 − 3x

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as \( x \to 3^- \).
2. Substitute \( x = 3 \) into the expression \( \frac{x - 4}{x^2 - 3x} \) to check for indeterminate form.
3. Calculate \( x^2 - 3x \) at \( x = 3 \) to see if it results in zero, indicating a potential vertical asymptote.
4. Factor the denominator \( x^2 - 3x \) as \( x(x - 3) \) to simplify the expression.
5. Analyze the behavior of the function as \( x \to 3^- \) by considering values of \( x \) slightly less than 3.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches 3 from the left, which is denoted as x→3⁻.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is crucial when evaluating limits, as discontinuities can lead to undefined behavior. In this problem, we need to check if the function is continuous at x = 3 to determine the limit accurately.

Factoring and Simplifying Expressions

Factoring and simplifying expressions is a key technique in calculus for evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. By factoring the numerator and denominator, we can often cancel common terms, making it easier to compute the limit. In this case, simplifying the expression will help us find the limit as x approaches 3.