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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.R.41
Chapter 2, Problem 2.R.41

Determine the following limits.
lim x→∞ (3 tan-1 x + 2)

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1
Recognize that the problem involves finding the limit of a function as \( x \) approaches infinity: \( \lim_{x \to \infty} (3 \tan^{-1} x + 2) \).
Recall the behavior of the inverse tangent function, \( \tan^{-1} x \), as \( x \to \infty \). The function approaches \( \frac{\pi}{2} \).
Substitute the asymptotic value of \( \tan^{-1} x \) into the expression: \( 3 \tan^{-1} x + 2 \approx 3 \left( \frac{\pi}{2} \right) + 2 \).
Simplify the expression by multiplying and adding the constants: \( 3 \times \frac{\pi}{2} + 2 \).
Conclude that the limit is the simplified expression from the previous step.

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Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine their end behavior. In this context, we analyze how the function approaches a specific value as x becomes very large.
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Inverse Tangent Function

The inverse tangent function, denoted as tan<sup>-1</sup>(x) or arctan(x), is a function that returns the angle whose tangent is x. As x approaches infinity, the value of arctan(x) approaches π/2. This property is essential for solving the limit in the question, as it helps us understand the limiting behavior of the function involved.
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Inverse Tangent

Constant Addition in Limits

When evaluating limits, adding a constant to a function does not affect the limit itself. This principle allows us to simplify the limit calculation by focusing on the behavior of the variable part of the function. In this case, after determining the limit of the arctan function, we can simply add 2 to find the final limit value.
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Related Practice
Textbook Question

Estimate the following limits using graphs or tables.


lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Textbook Question

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

Textbook Question

Determine the following limits.

lim x→∞ (5 + (cos4 x) / (x2 + x + 1))

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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 colony of squirrels is given by p(t)=15003+2e−0.1tp\(\left\)(t\(\right\))=\(\frac{1500}{3+2e^{-0.1t}\)}.

Textbook Question

Use the graph of ff in the figure to determine the values of xx in the interval (−3,5)\(\left\)(-3,5\(\right\)) at which f fails to be continuous. Justify your answers using the continuity checklist.

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Textbook Question

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

h(x)=2xx3−25xh\(\left\)(x\(\right\))=\(\frac{2x}{x^3-25x}\)

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