Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x2 − 25).
lim x → -5- f(x)
1. Limits and Continuity
Finding Limits Algebraically
5:50 minutesProblem 2.7.59
Textbook Question
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^- 1 / 1 − x=∞
Verified step by step guidance1
Understand the problem: We need to prove that the limit of the function \( \frac{1}{1-x} \) as \( x \) approaches 1 from the left (denoted as \( x \to 1^- \)) is infinity.
Consider the behavior of the function \( \frac{1}{1-x} \) as \( x \) approaches 1 from the left. As \( x \) gets closer to 1 from values less than 1, the denominator \( 1-x \) becomes a very small positive number.
Recognize that as \( 1-x \) approaches zero from the positive side, the fraction \( \frac{1}{1-x} \) becomes very large, since dividing by a smaller and smaller positive number results in a larger and larger value.
Formally, for any large positive number \( M \), we need to find a \( \delta > 0 \) such that if \( 0 < 1-x < \delta \), then \( \frac{1}{1-x} > M \). This is the definition of the limit approaching infinity.
Choose \( \delta = \frac{1}{M} \). Then, if \( 0 < 1-x < \delta \), it follows that \( \frac{1}{1-x} > M \), thus proving that \( \lim_{x \to 1^-} \frac{1}{1-x} = \infty \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Limits
Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. In this case, as x approaches 1 from the left, the function 1 / (1 - x) tends to infinity, indicating that the values of the function grow larger and larger.
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One-Sided Limits
One-Sided Limits
One-sided limits evaluate the behavior of a function as it approaches a specific point from one direction only. The notation lim x→1^- indicates that we are considering values of x that are less than 1, which is crucial for understanding how the function behaves as it nears the point of interest.
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Continuity and Discontinuity
Continuity refers to a function being unbroken and having no gaps at a point. In this case, the function 1 / (1 - x) is discontinuous at x = 1, as it approaches infinity from the left. Understanding this concept helps clarify why the limit diverges rather than converges to a finite value.
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