Evaluate each limit and justify your answer.
lim x→∞(2x+1x / x)^3

Verified step by step guidance

Step 1: Simplify the expression inside the limit. Start by rewriting the expression (2x + 1x) / x as (2x + x) / x.

Step 2: Simplify the expression further. Combine like terms in the numerator to get 3x / x.

Step 3: Simplify the fraction 3x / x. Since x is not zero, this simplifies to 3.

Step 4: Substitute the simplified expression back into the limit. The limit becomes lim x→∞ (3)^3.

Step 5: Evaluate the limit. Since 3 is a constant, the limit of a constant as x approaches infinity is simply the constant itself. Therefore, the limit is 3^3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, which can often simplify complex expressions. In this case, we analyze how the terms in the expression grow relative to each other as x becomes infinitely large.

Polynomial Growth

Polynomial growth refers to how polynomial functions behave as their variable approaches infinity. In the expression given, the highest degree term dominates the behavior of the function. Recognizing which terms are significant in the limit helps in simplifying the expression to find the limit more easily.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form, which can make evaluating limits more straightforward. In this limit problem, simplifying the expression before taking the limit allows for easier computation and clearer insight into the function's behavior as x approaches infinity.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{1}{2 x^{4} - \sqrt{4 x^{8} - 9 x^{4}}}$

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Textbook Question

If a function f represents a system that varies in time, the existence of lim ${$\displaystyle\[\lim$_{t$\rightarrow\]\infty$}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.