Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.
_____
√x² + 4
c. g(x) = -----------
x

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Step 1: Identify the function g(x) = (√(x² + 4)) / x. We need to find the horizontal asymptotes by evaluating the limits of g(x) as x approaches infinity and negative infinity.

Step 2: Consider the limit as x approaches infinity. Simplify the expression by dividing the numerator and the denominator by x, the highest power of x in the denominator: g(x) = (√(x² + 4) / x) = √(1 + 4/x²).

Step 3: Evaluate the limit as x approaches infinity: lim (x -> ∞) √(1 + 4/x²). As x becomes very large, 4/x² approaches 0, so the expression simplifies to √1 = 1. Therefore, the horizontal asymptote as x approaches infinity is y = 1.

Step 4: Consider the limit as x approaches negative infinity. The expression remains the same: g(x) = √(1 + 4/x²).

Step 5: Evaluate the limit as x approaches negative infinity: lim (x -> -∞) √(1 + 4/x²). Similarly, as x becomes very large in magnitude, 4/x² approaches 0, so the expression simplifies to √1 = 1. Therefore, the horizontal asymptote as x approaches negative infinity is also y = 1.

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주요 개념

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for analyzing the behavior of functions at specific points, particularly at infinity, which is crucial for determining asymptotes.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. A function has a horizontal asymptote if the limit of the function approaches a constant value as x approaches infinity, indicating that the function levels off at that value.

Rational Functions

Rational functions are ratios of two polynomials. Understanding their structure is vital for finding asymptotes, as the degrees of the numerator and denominator determine the existence and location of horizontal asymptotes, influencing the function's end behavior.

Horizontal and Vertical AsymptotesUse limits to determine the equations for all horizontal asymptotes._____√x² + 4c. g(x) = -----------x