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Chapter 2: Problem 79

Find each limit, if it exists. $$\lim _{x \rightarrow-\infty} \frac{2 x^{4}+x}{x+1}$$

Short Answer

Expert verified
The limit does not exist since the expression tends to \(-\text{\infty}\)

Step by step solution

01

Identify the highest degree term in the numerator and denominator

Look at the degrees of the polynomials in the numerator and the denominator. The highest degree term in the numerator is \(2x^4\) and in the denominator, it is \(x\).
02

Factor out the highest degree term from both numerator and denominator

Factor out \(x^4\) from the numerator and \(x\) from the denominator:\[\frac{2x^4 + x}{x + 1} = \frac{x^4 (2 + \frac{1}{x^3})}{x (1 + \frac{1}{x})}\]
03

Simplify the expression

Simplify the fraction:\[\frac{x^4 (2 + \frac{1}{x^3})}{x (1 + \frac{1}{x})} = x^3 \frac{2 + \frac{1}{x^3}}{1 + \frac{1}{x}}\]
04

Evaluate the limit

As \(x\) approaches \(-\infty\), \ \frac{1}{x^3} and \ \frac{1}{x} approach zero. So the limit becomes:\[\frac{2 + \frac{1}{x^3}}{1 + \frac{1}{x}} \rightarrow \frac{2 + 0}{1 + 0} = 2.\]Thus,\[\boxed{\text{lim}_{x \rightarrow -\text{\textbackslash \textbackslash \textinfty}} \frac{2x^4 + x}{x + 1} = -\text{\textbackslash \textbackslash \textinfty}}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits of Functions
In calculus, a limit helps us understand the behavior of a function as the input approaches a certain value. To calculate the limit of a function as x approaches a value, observe how the output values change.
Finding limits is crucial for defining derivatives and integrals.
Some common steps in finding limits include:
  • Identifying the highest degree terms
  • Simplifying fractions
  • Factoring expressions
In our example, we focus on the limit as x approaches \(-\infty\). This requires understanding the behavior of polynomials as x becomes indefinitely large in the negative direction.
Polynomial Functions
A polynomial function is composed of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. For instance, in the polynomial: $$2x^4 + x$$, $$2x^4$$ is the highest degree term. The degree of a polynomial dictates its growth rate.
For our problem, the degrees are:
  • Numerator: 4 (from $$2x^4$$)
  • Denominator: 1 (from $$x$$)
To simplify, we factor these values out, reducing our limit calculation to a simpler form and elucidating the dominant behavior of the function.
Asymptotic Behavior
Asymptotic behavior refers to how a function behaves as the input grows very large, either positively or negatively. This behavior is often revealed through limits. For large values, terms with lower powers become negligible.
In our example, as x approaches \(-\infty\):
The parts involving 1/x and 1/x^3 tend towards zero, simplifying our function. The limit hence simplifies to a constant term, reflecting the function's behavior at extreme values.
Understanding this concept helps in finding horizontal asymptotes and predicting the long-term behavior of functions.

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