Use the precise definition of infinite limits to prove the following limits.lim⁡x→−11(x+1)4=∞{\displaystyle\lim_{x\to-1}}\frac{1}{\left(x+1\$$right)^4$$}=\infty

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.7.46

Capitolo 2, Problema 2.7.46

Use the precise definition of infinite limits to prove the following limits.

$\lim_{x \to - 1} \frac{1}{{(x + 1)}^{4}} = \infty$

Guida verificata passo dopo passo

Step 1: Understand the definition of an infinite limit. The statement $ \lim_{x \to -1} \frac{1}{(x+1)^4} = \infty $ means that for every positive number $ M $, there exists a $ \delta > 0 $ such that if $ 0 < |x + 1| < \delta $, then $ \frac{1}{(x+1)^4} > M $.

Step 2: Analyze the function $ \frac{1}{(x+1)^4} $. As $ x $ approaches $ -1 $, the expression $ (x+1)^4 $ approaches 0, making $ \frac{1}{(x+1)^4} $ grow larger without bound.

Step 3: Set up the inequality $ \frac{1}{(x+1)^4} > M $ to find $ \delta $. This inequality can be rewritten as $ (x+1)^4 < \frac{1}{M} $.

Step 4: Solve $ (x+1)^4 < \frac{1}{M} $ for $ |x+1| $. Take the fourth root of both sides to get $ |x+1| < \frac{1}{M^{1/4}} $.

Step 5: Conclude that for any $ M > 0 $, choosing $ \delta = \frac{1}{M^{1/4}} $ ensures that $ 0 < |x+1| < \delta $ implies $ \frac{1}{(x+1)^4} > M $, thus proving the limit.

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Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value, leading the output to grow without bound. Specifically, if the limit of a function as x approaches a value c is infinity, it indicates that the function's values increase indefinitely as x gets closer to c. This concept is crucial for understanding how functions behave near vertical asymptotes.

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One-Sided Limits

Limit Definition

The precise definition of a limit involves the formal epsilon-delta approach, which provides a rigorous way to describe how a function behaves as it approaches a specific point. For infinite limits, this means that for every large number M, there exists a delta such that if the distance between x and c is less than delta, the function's value exceeds M. This definition is essential for proving limits rigorously.

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Polynomial Behavior Near Roots

Understanding how polynomials behave near their roots is vital for analyzing limits. In the case of the limit in question, the expression (x + 1)^4 approaches zero as x approaches -1, causing the overall fraction to approach infinity. Recognizing that higher powers of polynomials lead to faster growth or decay helps in predicting the behavior of functions near critical points.

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