Textbook Question
Determine the following limits.
b. lim x→4^− x − 5 / (x − 4)^2
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1. Limits and Continuity
Finding Limits Algebraically
2:53 minutesProblem 25a
Textbook Question
Determine the following limits.
a. lim z→3^+ (z − 1)(z − 2) / (z − 3)
Verified step by step guidance1
Step 1: Identify the type of limit problem. This is a one-sided limit as z approaches 3 from the right (z → 3^+).
Step 2: Analyze the behavior of the function as z approaches 3 from the right. The numerator (z - 1)(z - 2) is a polynomial that is continuous everywhere, so it will approach a finite value as z approaches 3.
Step 3: Consider the denominator (z - 3). As z approaches 3 from the right, (z - 3) approaches 0 from the positive side, which means the denominator is approaching zero.
Step 4: Determine the overall behavior of the function. Since the numerator approaches a finite value and the denominator approaches zero from the positive side, the limit will tend towards positive or negative infinity.
Step 5: Conclude the behavior of the limit. Since both (z - 1) and (z - 2) are positive when z is slightly greater than 3, the numerator is positive, and the denominator is positive, leading the limit to approach positive infinity.
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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 where they may not be defined. In this case, we are interested in the limit as z approaches 3 from the right (3+), which indicates we are looking at values of z that are slightly greater than 3.
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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 z→3^+ indicates that we are considering the limit as z approaches 3 from the right. This is crucial for understanding the behavior of functions that may have different values or undefined points at the limit point.
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Factoring and Simplifying Expressions
Factoring and simplifying expressions is a key technique in calculus for evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, the expression (z − 1)(z − 2) / (z − 3) can be analyzed by substituting values close to 3 to determine the limit's value, or by simplifying the expression if possible to avoid division by zero.
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