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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 2.7.48
Chapter 2, Problem 2.7.48

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


limx→0(1x4−sin(x))=∞{\(\displaystyle\[\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^4}\)-\(\sin\]\left\)(x\(\right\))\(\right\))=\(\infty\)

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Step 1: Understand the definition of an infinite limit. The limit \( \lim_{x \to a} f(x) = \infty \) means that for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( f(x) > M \).
Step 2: Consider the function \( f(x) = \frac{1}{x^4} - \sin(x) \). As \( x \to 0 \), \( \frac{1}{x^4} \to \infty \) and \( \sin(x) \to 0 \). Therefore, \( f(x) \to \infty \).
Step 3: For a given \( M > 0 \), we need to find \( \delta > 0 \) such that if \( 0 < |x| < \delta \), then \( \frac{1}{x^4} - \sin(x) > M \).
Step 4: Since \( \sin(x) \) is bounded, \( |\sin(x)| \leq |x| \). Therefore, \( \frac{1}{x^4} - \sin(x) \geq \frac{1}{x^4} - |x| \).
Step 5: Choose \( \delta \) small enough such that \( \frac{1}{x^4} - |x| > M \) for \( 0 < |x| < \delta \). This ensures that \( \frac{1}{x^4} - \sin(x) > M \), proving the limit is infinite.

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Key Concepts

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

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value, where the function's output grows without bound. Specifically, if the limit of a function as x approaches a value results in infinity, it indicates that the function increases indefinitely in that vicinity. Understanding infinite limits is crucial for analyzing functions that exhibit vertical asymptotes or unbounded growth.
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Sine Function Behavior

The sine function, denoted as sin(x), oscillates between -1 and 1 for all real numbers x. As x approaches 0, sin(x) can be approximated by its Taylor series expansion, which shows that sin(x) behaves like x near zero. This property is essential for evaluating limits involving sin(x) and understanding how it interacts with other functions, particularly in the context of limits approaching zero.
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Graph of Sine and Cosine Function

Limit Definition and Evaluation

The precise definition of a limit involves the concept of approaching a value as closely as desired. To prove limits, one often uses algebraic manipulation, substitution, or the epsilon-delta definition. In this case, evaluating the limit requires analyzing the expression as x approaches 0, particularly focusing on the dominant term in the expression, which is crucial for determining the limit's behavior.
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Related Practice
Textbook Question

Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.


g(x)=∣x+4x2−4∣g\(\left\)(x\(\right\))=\(\left\[\vert\]\frac{x+4}{x^2-4}\[\right\]\vert\)

Textbook Question

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→0 x^2=0 (Hint: Use the identity √x2=|x|.)

Textbook Question

Determine the following limits at infinity.


lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

Textbook Question

a. Analyze limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}} for each function.


f(x)=1+x−2x2−x3x2+1f\(\left\)(x\(\right\))=\(\frac{1+x-2x^2-x^3}{x^2+1}\)

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Textbook Question

Determine the following limits.


limθ→π2+13tanθ{\(\displaystyle\[\lim\)_{\(\theta\]\to\[\frac{\pi}{2}\)^{+}}\(\frac\)13\(\tan\]\theta\)}

Textbook Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

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