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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 2.5.61
Chapter 2, Problem 2.5.61

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 
f(x)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

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1
Step 1: Understand the function f(x) = sin(x). The sine function is periodic with a period of 2π, meaning it repeats its values every 2π units along the x-axis.
Step 2: Analyze the range of the sine function. The sine function oscillates between -1 and 1 for all x, so its range is [-1, 1].
Step 3: Determine the end behavior by considering the limits as x approaches positive and negative infinity. Since sin(x) is periodic and bounded, it does not approach a specific value as x approaches infinity or negative infinity.
Step 4: Identify any asymptotes. The sine function does not have any vertical or horizontal asymptotes because it is bounded and periodic.
Step 5: Sketch the graph of f(x) = sin(x). Draw a wave-like pattern oscillating between -1 and 1, repeating every 2π along the x-axis, with no asymptotes.

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Key Concepts

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

End Behavior of Functions

End behavior refers to the behavior of a function as the input values approach positive or negative infinity. Understanding end behavior is crucial for analyzing limits and determining how a function behaves at its extremes. For example, knowing whether a function approaches a specific value, diverges, or oscillates helps in sketching its graph accurately.
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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In the context of transcendental functions like sine, limits help determine the function's behavior at infinity, which is vital for analyzing end behavior.
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Asymptotes

Asymptotes are lines that a graph approaches but never touches, indicating the behavior of a function as it extends towards infinity. They can be vertical, horizontal, or oblique, depending on the function's characteristics. Identifying asymptotes is important for sketching graphs accurately, especially for functions that exhibit unbounded behavior or oscillation.
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Related Practice
Textbook Question

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

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Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

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Textbook Question

Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^+ 1 /1 − x=−∞

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Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

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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)=x+1x3−4x2+4xf\(\left\)(x\(\right\))=\(\frac{x+1}{x^3-4x^2+4x}\)

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=1−lnxf\(\left\)(x\(\right\))=1-\(\ln\) x

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