Textbook Question
Determine the following limits.
lim u→0^+ u − 1 / sin u
1. Limits and Continuity
Finding Limits Algebraically
1:37 minutesProblem 2.4.7a
Textbook Question
The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→1^− f(x)
Verified step by step guidance1
Identify that the limit \( \lim_{x \to 1^-} f(x) \) involves approaching the point \( x = 1 \) from the left side.
Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1, \( f(x) \) tends to either positive or negative infinity.
Since we are approaching from the left (\( x \to 1^- \)), examine the behavior of \( f(x) \) as \( x \) gets closer to 1 from values less than 1.
Consider the graph's trend as \( x \to 1^- \): if \( f(x) \) increases without bound, the limit is positive infinity; if it decreases without bound, the limit is negative infinity.
Conclude the analysis by determining the direction of \( f(x) \) as \( x \to 1^- \) based on the graph's behavior.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Asymptotes
Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.
Recommended video:
3:40
Introduction to Cotangent Graph Example 1
Limits
A limit describes the behavior of a function as the input approaches a particular point. The notation lim x→1^− f(x) specifically refers to the limit of f(x) as x approaches 1 from the left side. Understanding limits is crucial for analyzing the behavior of functions near points of discontinuity, such as vertical asymptotes.
Recommended video:
05:50One-Sided Limits
One-Sided Limits
One-sided limits evaluate the behavior of a function as the input approaches a specific point from one direction only. The notation lim x→1^− f(x) indicates that we are interested in the limit as x approaches 1 from values less than 1. This concept is essential for understanding how functions behave near points of discontinuity, particularly when vertical asymptotes are present.
Recommended video:
05:50One-Sided Limits
5:21mWatch next
Master Finding Limits by Direct Substitution with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice