Textbook Question
Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find
c. limx→4 (g(x))²
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1. Limits and Continuity
Finding Limits Algebraically
2:42 minutesProblem 2.4.18a
Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→1+ (√2x (x − 1)) / |x − 1|
Verified step by step guidance1
Identify the type of limit: This is a one-sided limit as x approaches 1 from the right (x → 1+). This means we are interested in values of x that are slightly greater than 1.
Analyze the expression: The expression is (√2x (x − 1)) / |x − 1|. Notice that the absolute value |x − 1| affects the expression differently depending on whether x is greater than or less than 1.
Simplify the expression for x > 1: Since x is approaching 1 from the right, x > 1, and thus |x − 1| = x − 1. Substitute this into the expression to simplify it.
Substitute and simplify: The expression becomes (√2x (x − 1)) / (x − 1). Cancel out the (x − 1) terms in the numerator and denominator, assuming x ≠ 1.
Evaluate the limit: After canceling, you are left with √2x. Now, substitute x = 1 into this simplified expression to find the limit as x approaches 1 from the right.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (denoted as lim x→c-) or the right (denoted as lim x→c+). Understanding one-sided limits is crucial for analyzing functions that may behave differently on either side of a point, especially at points of discontinuity.
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Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. In limit problems, it is important to consider how the absolute value affects the function's behavior, particularly when the input approaches a point where the expression inside the absolute value changes sign, as it can lead to different limit values.
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Limit Evaluation Techniques
Limit evaluation techniques involve various algebraic methods to find the limit of a function as it approaches a certain point. Techniques such as factoring, rationalizing, or applying L'Hôpital's Rule are often used to simplify expressions and resolve indeterminate forms, making it easier to compute the limit accurately.
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