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Magazine Chapter 2: Problem 17
Determine the infinite limit. $$ \lim _{x \rightarrow 1^{-}} \ln (1-|x|) $$
Short Answer
Expert verified
The limit is \(-\infty\).
Step by step solution
01
Understand the Function
We are given the function \(f(x) = \ln(1 - |x|)\). The goal is to find the limit of this function as \(x\) approaches 1 from the left, denoted \(x \to 1^{-}\).
02
Analyze the Absolute Value Expression
Because we are taking the limit as \(x\) approaches 1 from the left, \(|x| = x\). So, the expression becomes \(1 - |x| = 1 - x\) for \(x < 1\).
03
Substitute Into the Logarithm
Substituting \(1 - x\) into the logarithm, we now have \(\ln(1 - x)\). We need to analyze the behavior of \(\ln(1-x)\) as \(x\) approaches 1 from the left.
04
Evaluate the Behavior of 1-x
As \(x\) approaches 1 from the left, \(1 - x\) approaches 0 from the positive side (i.e., 0^{+}). This means \(1 - x \to 0^+\).
05
Determine the Behavior of the Logarithm
The natural logarithm \(\ln(y)\) approaches \(-\infty\) as the argument \(y\) approaches 0 from the positive side. Therefore, \(\ln(1-x)\to -\infty\) as \(x\to 1^{-}.\)
06
Conclude the Limit
Combining the previous steps, as \(x\to 1^{-}\), the function \(\ln(1-x)\) descends indefinitely towards \(-\infty\). Hence, the limit is \(-\infty\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Natural Logarithm
The natural logarithm, often denoted as \( \ln(x) \), is a fundamental mathematical function commonly used in calculus and algebra. This logarithm is based on the constant \( e \), an irrational number approximately equal to 2.71828. The natural logarithm is the inverse of the exponential function \( e^x \). Here are a few things to remember about \( \ln(x) \):
- It is only defined for positive numbers. As \( x \) approaches zero from the positive side \( (0^+) \), \( \ln(x) \) tends towards negative infinity \( (-\infty) \).
- As \( x \) increases, \( \ln(x) \) increases without bound, albeit at a decreasing rate.
- The graph of \( \ln(x) \) has a vertical asymptote at \( x=0 \) and passes through the point \( (1, 0) \).
Evaluating Limits
Evaluating limits is a critical concept in calculus, helping to understand the behavior of functions as they approach specific points. When you evaluate a limit, you determine what value a function approaches as its input approaches a particular point, even if it doesn't actually reach that point. Here are some techniques for evaluating limits:
- Direct Substitution: If substituting the point into the function gives a real number, that's the limit. However, this is not always possible, especially with undefined points.
- Factoring: Some limits can be simplified by factoring expressions, cancelling terms, and then using direct substitution.
- L'H么pital's Rule: Used when limits approach indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), allowing differentiation of the numerator and denominator before re-evaluation.
- Squeeze Theorem & Special Limits: Useful for more complex expressions; sometimes involving trigonometric functions or expressions with zero asymptotes.
Absolute Value Function
The absolute value function, denoted as \(|x|\), determines the magnitude of a number without considering its sign. This function returns the non-negative value of \( x \), regardless of whether \( x \) is positive or negative. Here are important points about absolute values:
- Definition: \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).
- Effect on Equations: Absolute value can affect equations by ensuring that distances or magnitudes are always non-negative.
- Graph Behavior: The graph of \(|x|\) is V-shaped, with its vertex at the origin, significantly affecting how limits are evaluated around critical points.
