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Chapter 2: Problem 12

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{3+2 x+4 x^{2}}{x^{2}}$$

Short Answer

Expert verified
Answer: The limit of the given rational function as x approaches infinity is 4.

Step by step solution

01

Identify the highest power of x in the denominator

In this problem, the highest power of x present in the denominator is x^2.
02

Divide each term in the numerator and denominator by x^2

Dividing each term in the numerator and denominator by x^2, we get: $$\frac{\frac{3}{x^2}+\frac{2x}{x^2}+\frac{4x^2}{x^2}}{\frac{x^2}{x^2}}$$
03

Simplify the rational function

Simplifying the expression, we get: $$\frac{\frac{3}{x^2}+\frac{2}{x}+4}{1}$$
04

Consider the limit as x approaches infinity

Now, we will evaluate the limit of the simplified expression as x approaches infinity: $$\lim _{x \rightarrow \infty} \frac{\frac{3}{x^2}+\frac{2}{x}+4}{1}$$ For each term with x in the denominator, as x approaches infinity, those terms will approach zero: $$\lim _{x \rightarrow \infty} \frac{3}{x^2}=0$$ $$\lim _{x \rightarrow \infty} \frac{2}{x}=0$$
05

Conclusion

Therefore, since the two terms with x in the denominator approach zero as x approaches infinity, the limit simplifies to: $$\lim _{x \rightarrow \infty} \frac{0+0+4}{1}=\lim _{x \rightarrow \infty} 4=4$$ Hence, the limit of the given rational function is 4.

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Most popular questions from this chapter

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x-1}{x^{2 / 3}-1}$$

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x^{2}-4 x+3}{x-1}$$

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