Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { 5 x ^ { 3 } + 8 x ^ { 2 } } { 3 x ^ { 4 } - 16 x ^ { 2 } }$$

Short answer

The limit of the function as \(x\) approaches 0 is \(-\frac{1}{2}\).

Step-by-step solution

Graphical Interpretation

By graphing the function \(\frac { 5 x ^ { 3 } + 8 x ^ { 2 } } { 3 x ^ { 4 } - 16 x ^ { 2 } }\), one may observe the graph's behavior as it approaches \(x = 0\). A noticeable trend in the graph would suggest the limit value.

Algebraic Evaluation - Factoring

First, factor out \(x^2\) from both the numerator and denominator of the function: \( \lim _ { x \rightarrow 0 } \frac { x^2(5x + 8) } { x^2(3x^2 - 16) }\). Here, you can observe a common factor of \(x^2\) in the numerator and denominator.

Algebraic Evaluation - Simplifying

Cancel out the common factor of \(x^2\) to simplify the function to \(\lim _ { x \rightarrow 0 } \frac {5x + 8} {3x^2 - 16 }\). Now, one can directly substitute \(x = 0\) into the function:Substituting \(x = 0\), we have: \(\frac {5(0) + 8} {3(0)^2 - 16 } = \frac {8}{-16} = -\frac{1}{2}\).

Conclusion

The limit of the function as \(x\) approaches 0 is \(-\frac{1}{2}\), as determined algebraically, and this should be confirmed by a close observation of the function's graph around \(x = 0\).

Key concepts

Graphical Limit Evaluation

Understanding the graphical evaluation of a limit is a foundational skill when exploring the behavior of functions. Imagine plotting the function on a graph: you will see its shape and how it behaves as it gets closer and closer to a certain point, which in this exercise is where x approaches 0. By carefully looking at the graph as x gets infinitesimally close to 0, we can infer the function's value at that point.

For the function \( \frac { 5x^3 + 8x^2 } { 3x^4 - 16x^2 } \) as x approaches 0, the graphical assessment will show us that the value of the function approaches a specific number, which is the limit. This visual observation is crucial as it not only offers a concrete understanding but also can act as a verification for our algebraic conclusion.

Algebraic Limit Confirmation

The algebraic confirmation of a limit requires manipulating the function with algebraic techniques to conclude the limit value. It provides mathematical proof that supports our graphical understanding. When we work out the problem algebraically, we substitute the value of x that it approaches into the simplified function. In our example,
\[ \lim _ { x \rightarrow 0 } \frac {5x + 8} {3x^2 - 16 } \],
after simplification and substitution of x with 0, we get \(-\frac{1}{2}\). This is an important step because it confirms the function's limit without any ambiguity or misinterpretation that may arise from graphical representation alone.

Factoring Algebraic Expressions

Factoring is a method of breaking down complex algebraic expressions into simpler components that are easier to manage. When we factor the expression
\[ \frac { 5x^3 + 8x^2 } { 3x^4 - 16x^2 } \]
the goal is to identify and extract any common factors in the numerator and denominator. In this specific case, we take out an x^2 from both to simplify the expression. Factoring is like breaking apart a puzzle into pieces that make sense individually and then rearranging them to form a clearer picture. This process can often reveal factors that can be canceled out, thus simplifying the expression and allowing us to find the limit without any indeterminate forms.

Simplifying Rational Expressions

When we simplify rational expressions, we're making the expression easier to understand and evaluate. The approach is to eliminate any common factors between the numerator and the denominator. Removing these factors, as we do when we cancel out the x^2 in our exercise, streamlines the expression. Simplification can reveal the true nature of the expression's behavior as it approaches a certain value for x.

Always remember that simplifying is not just about making the expression look easier; it's also about refining the equation to accurately assess limits and function behavior without unnecessary complications.