Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.7.41

Chapter 2, Problem 2.7.41

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→1 x^4=1

Verified step by step guidance

Step 1: Recall the precise definition of a limit. For the limit $ \lim_{{x \to 1}} x^4 = 1 $ to hold, for every $ \varepsilon > 0 $, there must exist a $ \delta > 0 $ such that whenever $ 0 < |x - 1| < \delta $, it follows that $ |x^4 - 1| < \varepsilon $.

Step 2: Start by expressing $ |x^4 - 1| $ in a form that can be manipulated. Notice that $ x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) $.

Step 3: To ensure $ |x^4 - 1| < \varepsilon $, we need to bound $ |(x - 1)(x + 1)(x^2 + 1)| $. Assume $ |x - 1| < 1 $, which implies $ 0 < x < 2 $. This gives us bounds for $ |x + 1| $ and $ |x^2 + 1| $.

Step 4: Calculate the bounds: $ |x + 1| < 3 $ and $ |x^2 + 1| < 5 $ when $ 0 < x < 2 $. Therefore, $ |(x - 1)(x + 1)(x^2 + 1)| < |x - 1| \cdot 3 \cdot 5 = 15|x - 1| $.

Step 5: To satisfy $ |x^4 - 1| < \varepsilon $, choose $ \delta = \frac{\varepsilon}{15} $. This ensures that whenever $ 0 < |x - 1| < \delta $, it follows that $ |x^4 - 1| < \varepsilon $, thus proving the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formalism is crucial for proving limits rigorously.

Epsilon-Delta Relationship

In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to c. Establishing a relationship between ε and δ is essential to demonstrate that as x gets sufficiently close to c, f(x) will be within ε of L.

Polynomial Functions

Polynomial functions, such as f(x) = x^4, are continuous everywhere on their domain. This property simplifies limit calculations, as the limit of a polynomial as x approaches a point can be found by direct substitution, making it easier to apply the ε-δ definition.

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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.lim x→1 x^4=1

Verified video answer for a similar problem:

Key Concepts

Limit Definition

Epsilon-Delta Relationship

Polynomial Functions

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→1 x^4=1