Textbook Question
Determine the following limits.
lim x→∞ (3x12 − 9x7)
1. Limits and Continuity
Finding Limits Algebraically
2:30 minutesProblem 70b
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).
f(x) = (x2 − 4x + 3) / (x − 1)
Verified step by step guidance1
Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: x - 1 = 0, which gives x = 1.
Step 2: Confirm that x = 1 is a vertical asymptote by checking that the numerator does not also become zero at x = 1. Substitute x = 1 into the numerator: (1)^2 - 4(1) + 3 = 0, which means the numerator is zero, so x = 1 is not a vertical asymptote but a removable discontinuity.
Step 3: Factor the numerator to simplify the expression. The numerator x^2 - 4x + 3 can be factored as (x - 1)(x - 3).
Step 4: Simplify the function by canceling the common factor (x - 1) from the numerator and the denominator, resulting in f(x) = x - 3 for x ≠ 1.
Step 5: Since the factor (x - 1) cancels out, there are no vertical asymptotes. However, analyze the behavior around x = 1 to understand the nature of the discontinuity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Asymptotes
Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator does not. These points indicate where the function's value tends to infinity or negative infinity. To find vertical asymptotes, set the denominator equal to zero and solve for x, identifying the values that make the function undefined.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, evaluating the left-hand limit (lim x→a<sup>-</sup> f(x)) and the right-hand limit (lim x→a<sup>+</sup> f(x)) helps determine the behavior of the function near the asymptote, indicating whether it approaches positive or negative infinity.
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Rational Functions
Rational functions are ratios of polynomials, expressed as f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of these functions, particularly their asymptotic behavior, is influenced by the degrees of the polynomials in the numerator and denominator. Understanding the structure of rational functions is crucial for analyzing their limits and identifying asymptotes.
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