Question

Determine the following limits. $$\lim _{x \rightarrow-\infty} \frac{40 x^{4}+x^{2}+5 x}{\sqrt{64 x^{8}+x^{6}}}$$

Short answer

Answer: The limit of the given function as x approaches negative infinity is 5.

Step-by-step solution

Identify the dominating term in the numerator and denominator

The dominating term in the numerator is the term with the highest degree, which is 40x^4. In the denominator, the dominating term is the square root of 64x^8.

Divide all terms in the expression by the dominating term

We will divide the entire expression by x^4 in the numerator and sqrt(x^8) in the denominator. $$\lim_{x \to -\infty} \frac{40x^4+x^2+5x}{\sqrt{64x^8+x^6}} = \lim_{x \to -\infty} \frac{40 + \frac{x^2}{x^4} + \frac{5x}{x^4}}{\sqrt{ \frac{64x^8}{x^8} + \frac{x^6}{x^8}} }$$

Simplify the expression after dividing terms

Now, we will simplify the expression and evaluate the non-dominating terms as x approaches negative infinity. $$\lim_{x \to -\infty} \frac{40 + \frac{x^2}{x^4} + \frac{5x}{x^4}}{\sqrt{ \frac{64x^8}{x^8} + \frac{x^6}{x^8}} } = \lim_{x \to -\infty} \frac{40 + \frac{1}{x^2} + \frac{5}{x^3}}{\sqrt{ 64 + \frac{1}{x^2}}}$$

Evaluate the limit as x approaches negative infinity

As x approaches negative infinity, the non-dominating terms in the expression (1/x^2, 5/x^3) will approach zero, and the remaining terms will be the dominating terms. $$\lim_{x \to -\infty} \frac{40 + 0 + 0}{\sqrt{64 + 0}} = \frac{40}{\sqrt{64}}$$

Calculate the final result

Now, we will compute the final result: $$\frac{40}{\sqrt{64}} = \frac{40}{8} = 5$$ Therefore, the limit of the given function as x approaches negative infinity is 5.

Key concepts

Dominating Terms

In calculus, identifying the dominating terms of a function is crucial when evaluating limits. Dominating terms are essentially the terms in a mathematical expression that grow fastest or shrink slowest as the variable approaches infinity or any other point of interest.
The goal is to find the highest power of the variable, or the most significant component of a radical in both the numerator and the denominator. This is because these terms determine the overall behavior of the function as it becomes very large or very small.
For example, in the equation \( \frac{40x^4+x^2+5x}{\sqrt{64x^{8}+x^{6}}} \), the dominating term in the numerator is \( 40x^4 \) while in the denominator the dominating term is \( \sqrt{64x^8} \).
Recognizing these allows you to properly simplify expressions and focus on the terms that have the greatest impact on the limit behavior.

Simplifying Expressions

Once you've identified the dominating terms, the next step is simplifying the expression. The process of simplification in the context of limits often involves dividing all terms by the dominating term.
This helps in making the expression much easier to evaluate as other, less significant terms diminish to zero as the variable approaches either infinity or a point of finite value.
Consider the transformation of the expression \( \frac{40x^4+x^2+5x}{\sqrt{64x^8+x^6}} \) after dividing by \( x^4 \) in the numerator and \( \sqrt{x^8} \) in the denominator. This gives us \( \frac{40 + \frac{x^2}{x^4} + \frac{5x}{x^4}}{\sqrt{ \frac{64x^8}{x^8} + \frac{x^6}{x^8}}} \).
Through this simplification, complex expressions become manageable, allowing one to more easily evaluate the limit.

Evaluating Limits

After simplification, the core part is evaluating the actual limit. When x approaches negative infinity, the lesser terms (non-dominating terms) in the expression go to zero.
This makes it possible to disregard them and focus only on the previously identified dominating terms. In our example, terms like \( \frac{1}{x^2} \) and \( \frac{5}{x^3} \) approach zero because they contain x in the denominator.
By focusing on dominating terms, the simplified expression becomes \( \frac{40 + 0 + 0}{\sqrt{64 + 0}} \), resulting in \( \frac{40}{8} = 5 \).
Evaluating the limit is essentially discovering the end-behavior of the function as the variable extends towards infinity or any target value.

Asymptotic Behavior

Asymptotic behavior in limits gives insight into how a function behaves as it approaches a specific point or extends to infinity. Functions with a defined asymptote do not actually reach the asymptote but get indefinitely closer.
In calculus, asymptotic analysis helps in predicting this behavior by focusing on dominating terms and their influence on the overall function.
In the example \( \lim_{x \to -\infty} \frac{40x^4+x^2+5x}{\sqrt{64x^8+x^6}} \), the asymptotic behavior is derived from understanding that only the highest powers of x matter in extreme values.
Simplifying and evaluating the limit gives us the horizontal asymptote: \( y = 5 \). This means as x approaches negative infinity, the function approaches a line at height 5, without actually touching it.
Understanding asymptotes enables us to predict and describe the end-behavior of graphs and functions without complete calculation, which is very helpful in calculus and mathematics in general.