Calculate the limits in exercises, using only the continuity...

Calculate the limits in exercises, using only the continuity of linear and power functions and the limit rules. Cite each limit rule that you apply.
$$
\lim _{x \rightarrow-1} \frac{x-1}{(x+4)(x+2)}
$$

Key Concepts

Rational Functions

A rational function is a quotient of two polynomials. The limit of a rational function at a point where the denominator is not zero can be found by direct substitution, since both the numerator and denominator are continuous at that point.

Polynomial Functions

Polynomial functions, defined by expressions involving only powers of x with constant coefficients, are continuous everywhere. This means that when evaluating the limit of a polynomial as x approaches any value, the limit is attained simply by substituting the value into the polynomial.

Direct Substitution

Direct substitution is a method for evaluating limits where the variable is replaced by the value it is approaching. This method is valid when the function is continuous at that point, such as with linear, polynomial, and certain rational functions.

Continuity

Continuity is the property of a function where the limit as x approaches a point is equal to the function's value at that point. For many functions, including polynomials and simple rational functions (provided the denominator is not zero), continuity allows us to evaluate limits by direct substitution.

Limit Laws

The limit laws, including the sum, product, and quotient laws, are fundamental rules in calculus that allow one to break down complex limit expressions into simpler parts. By applying these laws, one can compute the limit of a function by separately evaluating the limits of its individual components.

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