Calculating Limits


Find the limits in Exercises 11–22.


limx→−1/2 4x(3x+4)²

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to -\frac{1}{2}} 4x(3x+4)^2 \).
2. Substitute \( x = -\frac{1}{2} \) directly into the expression to check if it results in a determinate form.
3. Calculate \( 3x + 4 \) by substituting \( x = -\frac{1}{2} \), which gives \( 3(-\frac{1}{2}) + 4 = -\frac{3}{2} + 4 = \frac{5}{2} \).
4. Substitute \( x = -\frac{1}{2} \) into \( 4x \), which gives \( 4(-\frac{1}{2}) = -2 \).
5. Combine the results: \( -2 \times (\frac{5}{2})^2 \) to find the limit. Simplify the expression to complete the calculation.

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 defining derivatives and integrals. In this case, we are interested in the limit of the function as x approaches -1/2.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function in the limit problem, 4x(3x+4)², is a polynomial function, and understanding its structure is essential for evaluating limits. Polynomial functions are continuous everywhere, which simplifies the process of finding limits.

Substitution Method

The substitution method is a technique used to evaluate limits by directly substituting the value that x approaches into the function, provided the function is continuous at that point. If direct substitution results in an indeterminate form, further algebraic manipulation may be necessary. In this case, substituting x = -1/2 into the polynomial will help find the limit.