Finding One-Sided Limits Algebraically


Find the limits in Exercises 11–20.


a. limx→1+ (√2x (x − 1)) / |x − 1|

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.