Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 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 specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors depending on the values of x, particularly at points where the denominator is zero. Understanding how to simplify and analyze these functions is essential for finding limits, especially when approaching points that may lead to indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require additional techniques, such as L'Hôpital's Rule or algebraic manipulation, to resolve and find the actual limit.