Question

Evaluate each limit. $$\lim _{x \rightarrow-1}\left(x^{3}-3 x^{2}-5 x-5\right)$$

Short answer

\(-4\)

Step-by-step solution

Direct Substitution

Attempt to evaluate the limit by directly substituting the value for which the variable is approaching into the function. Here, substitute \( x = -1 \) into the polynomial function \( x^3 - 3x^2 - 5x - 5 \).

Evaluate the Polynomial

Evaluate each term of the polynomial with \( x = -1 \): \( (-1)^3 - 3(-1)^2 - 5(-1) - 5 \). Simplify each term to get \( -1 - 3(1) + 5 - 5 \).

Combine Like Terms

Combine the terms to find the value of the polynomial at \( x = -1 \): \( -1 - 3 + 5 - 5 = -4 \)

Key concepts

Direct Substitution Method

The direct substitution method is a fundamental approach in calculus for evaluating limits. It involves replacing the variable in the function with the value it is approaching. This method is based on the idea that if a function is continuous at a point, the limit as the variable approaches that point will equal the function's value at that point.

To apply this method, simply substitute the approaching value into the function and simplify the expression. If the result is a real number, then that number is the limit of the function as the variable approaches the given value. However, if direct substitution leads to an undefined expression, such as division by zero or an indeterminate form, this method cannot be used and other limit evaluation techniques, such as factoring, conjugates, or L'Hôpital's Rule, may be necessary.

Evaluating Polynomials

Evaluating polynomials is a straightforward process when using direct substitution. A polynomial is an algebraic expression composed of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

When evaluating a polynomial at a specific value, each variable in the polynomial is replaced with that value. To ensure accuracy during the evaluation, follow the standard order of operations: first calculate the exponents, then perform any multiplication or division from left to right, and finally carry out addition and subtraction from left to right. Simplify each term separately before adding or subtracting them to find the overall value of the polynomial.

Combining Like Terms

Combining like terms is an essential simplification technique in algebra. Like terms are terms within an equation or expression that have exactly the same variables raised to the same powers, but possibly different coefficients.

In the process of simplifying an expression, after evaluating each term following a substitution, identify and group like terms together. Then, simply add or subtract the coefficients of these like terms. This step not only makes the expression neater but is also crucial when you are working to find the limit of a function at a certain point. Remember that only like terms can be combined; terms with different variables or exponents must remain separate.