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)

Guida verificata passo dopo passo

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.

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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.

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Percorso guidato

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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.

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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.

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One-Sided Limits

Pratica correlata

Domanda del libro di testo

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

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Domanda del libro di testo

Complete the following sentences in terms of a limit.

a. A function is continuous from the left at a if _____.