Find lim_(x rarr oo)(x+1)/(x^(2)+1). Choose 1 answer: (A) 1 (B) 2 (c) 0 (D) The limit is unbounded

Analyze Behavior Separately: To find the limit of the function as x approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.Degree of Polynomials: The degree of the polynomial in the denominator (x^2+1) is higher than the degree of the polynomial in the numerator (x+1). This means that as x becomes very large, the denominator will grow much faster than the numerator.Simplify Expression: We can divide both the numerator and the denominator by x^2, the highest power of x in the denominator, to simplify the expression and see the behavior as x approaches infinity. (x+1)/(x^2+1) = (1/x + 1/x^2) / (1 + 1/x^2)Approaching Infinity: As x approaches infinity, the terms 1/x and 1/x^2 in the numerator and the term 1/x^2 in the denominator approach 0. So, the expression simplifies to 0/1.Limit Calculation: Therefore, the limit of (x+1)/(x^2+1) as x approaches infinity is 0.

Home

Math Problems

Algebra 2

Add, subtract, multiply, and divide polynomials

Full solution

Q. Find

\lim _{x \rightarrow \infty} \frac{x+1}{x^{2}+1}

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

2

\newline

(C)

0

\newline

(D) The limit is unbounded

Analyze Behavior Separately: To find the limit of the function as $x$ approaches infinity, we need to analyze the behavior of the numerator and the denominator separately.
Degree of Polynomials: The degree of the polynomial in the denominator $x^2+1$ is higher than the degree of the polynomial in the numerator $x+1$ . This means that as $x$ becomes very large, the denominator will grow much faster than the numerator.
Simplify Expression: We can divide both the numerator and the denominator by $x^2$ , the highest power of $x$ in the denominator, to simplify the expression and see the behavior as $x$ approaches infinity. $\newline$ $\frac{x+1}{x^2+1} = \frac{\frac{1}{x} + \frac{1}{x^2}}{1 + \frac{1}{x^2}}$
Approaching Infinity: As $x$ approaches infinity, the terms $\frac{1}{x}$ and $\frac{1}{x^2}$ in the numerator and the term $\frac{1}{x^2}$ in the denominator approach $0$ . So, the expression simplifies to $\frac{0}{1}$ .
Limit Calculation: Therefore, the limit of $(x+1)/(x^2+1)$ as $x$ approaches $ext{infinity}$ is $0$ .