Question

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 1 / 2 } 3 x ^ { 2 } ( 2 x - 1 )$$

Short answer

The limit as \(x\) approaches \(-1/2\) of the function \(3x^{2}(2x-1)\) is \(-3/4\).

Step-by-step solution

Substitution

Substitute the limiting value \(-1/2\) into the function. This means replacing each \(x\) in the function with \(-1/2\), to get: \(3(-1/2)^2 (2(-1/2) - 1)\)

Simplify Each Part

Simplify each part of the expression we got from the substitution separately. \((-1/2)^2 = 1/4\), and \(2(-1/2) - 1 = -2-1 = -1\), so that we notice our equation now is, \(3*(1/4)*(-1)\)

Solve

Solve the simplified equation \(3*(1/4)*(-1)\) to get \(-3/4\). This is the limit of the function as \(x\) approaches \(-1/2\).

Key concepts

Limit by Substitution

Understanding the 'limit by substitution' is pivotal in the field of calculus. It's a technique applied when you need to find the limit of a function as the variable approaches a specific value. Let's say you have a function f(x) and you want to find the limit as x approaches the value 'a'. Instead of working with complex calculations, you often can simply substitute 'a' into f(x) to find the limit, provided that f(x) is continuous at point 'a'.

In our example, we are asked to find \(\lim _ {x \rightarrow -1 / 2} 3x^2(2x - 1)\). The substitution method involves replacing x with -1/2 directly in the expression. If the function is well-behaved at that point (that is, it's not undefined or does not lead to an indeterminate form like \(0/0\)), then the evaluated expression gives the limit immediately.

Simplifying Algebraic Expressions

The step involving the 'simplification of algebraic expressions' plays a significant role in not just understanding limits, but in all areas of algebra. Simplifying expressions helps to reveal the core part of the function that determines its behavior near the limit point.

When we are given an algebraic expression, like in the substituted expression \(3(-1/2)^2 (2(-1/2) - 1)\), our goal is to break it down to its simplest form. This means performing all the basic arithmetic operations: squaring \( -1/2 \), multiplying by 2, and then subtracting 1. Once we simplify these individual components, we can multiply them together to find the overall simplified form. In our case, simplifying the expression led us to \(3*(1/4)*(-1)\), a much more manageable equation to evaluate.

Graphical Representation of Limits

The 'graphical representation of limits' allows students to visualize the concept of a limit in a tangible way. By plotting the function on a graph, one can see how the function behaves as it nears the point in question.

Graphs can also provide insight into whether a function is continuous, if there are any asymptotes, or if the function behaves unexpectedly at certain points. For the exercise \(\lim _ {x \rightarrow -1 / 2} 3x^2(2x - 1)\), drawing the graph of the function would display its curve. If you trace the curve as x approaches -1/2, you’ll notice whether it approaches a specific y-value, which would be the limit. Graphical analysis is particularly useful when algebraic methods are difficult to apply, or when the limit leads to an indeterminate form and requires further insight.