Problem 5
Question
Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}+x-1}{x^{2}-x+4} $$
Step-by-Step Solution
Verified Answer
The limit is 2.
1Step 1: Identify the Dominant Term
When evaluating limits as \(x\) approaches infinity, focus on the highest power (degree) of \(x\) in both the numerator and the denominator. Here, the dominant term in both the numerator \(2x^2 + x - 1\) and the denominator \(x^2 - x + 4\) is \(x^2\).
2Step 2: Rewrite the Expression
To simplify the limit, divide every term in the numerator and the denominator by \(x^2\), the highest power of \(x\) in the expression.\[ \lim_{x \to \infty} \frac{2x^2 + x - 1}{x^2 - x + 4} = \lim_{x \to \infty} \frac{2 + \frac{1}{x} - \frac{1}{x^2}}{1 - \frac{1}{x} + \frac{4}{x^2}} \]
3Step 3: Evaluate the Limit
As \(x\) approaches infinity, the terms \(\frac{1}{x}\), \(\frac{1}{x^2}\), and \(\frac{4}{x^2}\) tend towards zero. Substituting these limits into the expression gives:\[\lim_{x \to \infty} \frac{2 + 0 - 0}{1 - 0 + 0} = \frac{2}{1} = 2\]
4Step 4: Conclude the Solution
The simplification and evaluation show that the original limit simplifies to 2. The dominant behavior of the polynomial function determines that the end-behavior for this rational function approaches a constant value.
Key Concepts
Dominant TermRational FunctionsInfinite Limits
Dominant Term
When tackling limits at infinity, the concept of the 'dominant term' is foundational. Understanding this can simplify your work immensely.
In any polynomial expression, the dominant term is the one with the highest power of \(x\). Here, this is because, as \(x\) grows infinitely large, the highest powers will "dominate" the behavior of the polynomial.
For example, consider the polynomial \(2x^2 + x - 1\). As \(x\) approaches infinity, \(x^2\) grows much faster than \(x\) or \(-1\), making \(x^2\) the dominant term. When simplifying rational functions like the one in this exercise, focusing on these terms can help make handling the limit more straightforward. To solve limits like \( \lim_{x \to \infty} \frac{2x^2 + x - 1}{x^2 - x + 4} \), identifying the dominant term in both the numerator and the denominator, here both \(x^2\), allows us to simplify by dividing through by \(x^2\). This step removes less impactful terms and isolates the key behavior of the function for large values of \(x\).
In any polynomial expression, the dominant term is the one with the highest power of \(x\). Here, this is because, as \(x\) grows infinitely large, the highest powers will "dominate" the behavior of the polynomial.
For example, consider the polynomial \(2x^2 + x - 1\). As \(x\) approaches infinity, \(x^2\) grows much faster than \(x\) or \(-1\), making \(x^2\) the dominant term. When simplifying rational functions like the one in this exercise, focusing on these terms can help make handling the limit more straightforward. To solve limits like \( \lim_{x \to \infty} \frac{2x^2 + x - 1}{x^2 - x + 4} \), identifying the dominant term in both the numerator and the denominator, here both \(x^2\), allows us to simplify by dividing through by \(x^2\). This step removes less impactful terms and isolates the key behavior of the function for large values of \(x\).
Rational Functions
Rational functions are expressions that involve ratios of polynomials. The general form is \(\frac{P(x)}{Q(x)}\) where both \(P(x)\) and \(Q(x)\) are polynomials. Evaluating limits for these functions, especially as \(x\to\infty\), can tell us about the function's end behavior.
One key thing to remember is that with rational functions, the dominant term you identified helps simplify both the numerator and the denominator. Taking the highest power term in both allows dividing the entire function by this term. This reduces complexities and allows easier evaluation of limits as it simplifies contributions from lesser degree terms to near zero as \(x\) becomes extremely large.
In practice, after identifying the dominant terms, rewrite the function. For instance:
One key thing to remember is that with rational functions, the dominant term you identified helps simplify both the numerator and the denominator. Taking the highest power term in both allows dividing the entire function by this term. This reduces complexities and allows easier evaluation of limits as it simplifies contributions from lesser degree terms to near zero as \(x\) becomes extremely large.
In practice, after identifying the dominant terms, rewrite the function. For instance:
- For the function \(\frac{2x^2 + x - 1}{x^2 - x + 4}\), divide each term by \(x^2\).
- This yields: \(\frac{2 + \frac{1}{x} - \frac{1}{x^2}}{1 - \frac{1}{x} + \frac{4}{x^2}}\).
Infinite Limits
Infinite limits help us understand the behavior of functions as values approach infinity, or in some cases, as they grow without bound. In the context of rational functions, this often concerns what the function's output approaches as its input grows larger and larger.
When evaluating these limits, particularly \(\lim_{x \rightarrow \infty}\), notice how smaller terms in the polynomial can tend towards zero. For example, terms like \(\frac{1}{x}\) or \(\frac{4}{x^2}\) become insignificant when compared to constant terms such as \(2\) or \(1\).
When evaluating these limits, particularly \(\lim_{x \rightarrow \infty}\), notice how smaller terms in the polynomial can tend towards zero. For example, terms like \(\frac{1}{x}\) or \(\frac{4}{x^2}\) become insignificant when compared to constant terms such as \(2\) or \(1\).
- This holds true because dividing a constant by increasingly large numbers results in a value getting closer to zero.
- Thus, in the expression \(\frac{2 + 0 - 0}{1 - 0 + 0}\), all smaller terms vanish, simplifying the limit to \(\frac{2}{1} = 2\).
Other exercises in this chapter
Problem 4
Find all antiderivatives of the given function. $$ -6 x+5 $$
View solution Problem 4
Find all critical numbers of the given function. $$ f(x)=x^{5}-5 x^{3}+10 x-3 $$
View solution Problem 5
Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ g(x)=\frac{x}{x^{2}+1} $$
View solution Problem 5
Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))
View solution