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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 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

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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.
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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.
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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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Related Practice
Textbook Question

Determine the interval(s) on which the following functions are continuous. 

f(x)=1 / x^2−4

Textbook Question

If a function f represents a system that varies in time, the existence of lim limt→∞f(t){\(\displaystyle\[\lim\)_{t\(\rightarrow\]\infty\)}{f(t)}} means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.


The population of a bacteria culture is given by p(t)=2500t+1p\(\left\)(t\(\right\))=\(\frac{2500}{t+1}\).

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Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

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Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x3+12x3+16x6+1f\(\left\)(x\(\right\))=\(\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}\)}

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Textbook Question

Find an interval containing a solution to the equation 2x=cos(x)2x=\(\cos\]\left\)(x\(\right\)). Use a graphing utility to approximate the solution.

Textbook Question

Find all vertical asymptotes x=ax=a of the following functions. For each value of aa, determine limx→a+f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{+}}}f\(\left\)(x\(\right\)), limx→a−f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{-}}}f\(\left\)(x\(\right\)), and limx→af(x){\(\displaystyle\]\lim\)_{x\(\to\) a}}f\(\left\)(x\(\right\)).

f(x)=cos(x)x2+2xf\(\left\)(x\(\right\))=\(\frac{\cos\left(x\right)}{x^2+2x}\)