Textbook Question
Which of the following statements are correct? Choose all that apply.
a. lim x→1 1/ (x−1)^2 does not exist
b. lim x→1 1/ (x−1)^2=∞
c. lim x→1 1/(x−1)^2=−∞
1. Limits and Continuity
Finding Limits Algebraically
1:58 minutesProblem 23b
Textbook Question
Determine the following limits.
b. lim x→4^− x − 5 / (x − 4)^2
Verified step by step guidance1
Step 1: Identify the type of limit problem. This is a one-sided limit as \( x \) approaches 4 from the left (denoted by \( x \to 4^- \)).
Step 2: Substitute \( x = 4 \) into the expression \( \frac{x - 5}{(x - 4)^2} \) to check for any indeterminate forms. You will find that the numerator becomes \( 4 - 5 = -1 \) and the denominator becomes \( (4 - 4)^2 = 0 \), indicating a division by zero, which suggests a potential vertical asymptote.
Step 3: Analyze the behavior of the function as \( x \) approaches 4 from the left. Since \( x \to 4^- \), \( x \) is slightly less than 4, making \( x - 4 \) a small negative number. Therefore, \( (x - 4)^2 \) is a small positive number.
Step 4: Consider the sign of the expression. The numerator \( x - 5 \) is negative when \( x \) is slightly less than 4, and the denominator \( (x - 4)^2 \) is positive. Thus, the overall expression \( \frac{x - 5}{(x - 4)^2} \) is negative.
Step 5: Conclude the behavior of the limit. As \( x \to 4^- \), the expression \( \frac{x - 5}{(x - 4)^2} \) approaches negative infinity because the numerator is negative and the denominator approaches zero from the positive side.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or where they are not defined. In this case, we are interested in the limit as x approaches 4 from the left (denoted as x→4^−).
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation x→4^− indicates that we are considering values of x that are less than 4. This is crucial for determining the limit in cases where the function may behave differently from the left and right of the point in question.
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Indeterminate Forms
Indeterminate forms occur in limit problems when direct substitution leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this problem, substituting x = 4 into the expression results in the form 0/0, indicating that further analysis, such as factoring or applying L'Hôpital's Rule, is necessary to evaluate the limit.
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