Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.


f(x)=√x^2+2x+6−3 / x−1

Verified step by step guidance

1. Step 1: Identify the dominant terms in the numerator and the denominator as x approaches infinity. For the function \( f(x) = \frac{\sqrt{x^2 + 2x + 6} - 3}{x - 1} \), the dominant term in the numerator is \( \sqrt{x^2} = x \) and in the denominator is \( x \).
2. Step 2: Simplify the expression by dividing both the numerator and the denominator by the dominant term \( x \). This gives \( \frac{\sqrt{x^2 + 2x + 6}/x - 3/x}{x/x - 1/x} \).
3. Step 3: Simplify further by recognizing that \( \sqrt{x^2 + 2x + 6}/x = \sqrt{1 + 2/x + 6/x^2} \). As \( x \to \infty \), \( 2/x \to 0 \) and \( 6/x^2 \to 0 \), so \( \sqrt{1 + 2/x + 6/x^2} \to 1 \).
4. Step 4: Evaluate the limit as \( x \to \infty \). The expression simplifies to \( \frac{1 - 0}{1 - 0} = 1 \). Therefore, \( \lim_{x \to \infty} f(x) = 1 \).
5. Step 5: Evaluate the limit as \( x \to -\infty \). The dominant term in the numerator becomes \( -x \) because \( \sqrt{x^2} = |x| \) and \( x \) is negative. Simplifying the expression similarly, we find \( \lim_{x \to -\infty} f(x) = -1 \). Thus, the horizontal asymptotes are \( y = 1 \) and \( y = -1 \).

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine the end behavior of the function, which is crucial for identifying horizontal asymptotes. For rational functions, this often involves comparing the degrees of the numerator and denominator.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input values become very large or very small. They indicate the value that the function approaches at infinity. To find horizontal asymptotes, one typically evaluates the limits of the function as x approaches positive and negative infinity.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions at infinity is influenced by the degrees of the polynomials in the numerator and denominator. Understanding how to simplify and analyze these functions is essential for determining limits and asymptotic behavior.