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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 2.39
Chapter 2, Problem 2.39

Estimate the following limits using graphs or tables.


lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

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1
Identify the limit expression: \( \lim_{{x \to 1}} \frac{9(\sqrt{2x} - x^4 - 3\sqrt{x})}{1 - x^{3/4}} \).
Recognize that direct substitution of \( x = 1 \) results in an indeterminate form \( \frac{0}{0} \).
Consider using a table of values to estimate the limit by choosing values of \( x \) that approach 1 from both the left and the right.
Alternatively, graph the function \( f(x) = \frac{9(\sqrt{2x} - x^4 - 3\sqrt{x})}{1 - x^{3/4}} \) and observe the behavior as \( x \) approaches 1.
Analyze the behavior of the numerator and the denominator separately as \( x \to 1 \) to understand the limit's behavior.

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Evaluating limits can involve direct substitution, factoring, or using special techniques like L'Hôpital's rule when dealing with indeterminate forms.
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Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or holes at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined or infinite limits.
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Graphical Analysis

Graphical analysis involves using the visual representation of a function to estimate limits and understand its behavior. By plotting the function, one can observe trends, identify asymptotes, and determine the value the function approaches as the input nears a specific point. This method is particularly useful for complex functions where algebraic manipulation may be challenging.
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Related Practice
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

Determine the following limits.


limx→0−csc(x){\(\displaystyle\[\lim\)_{x\(\to\)0^{-}}}\(\csc\]\left\)(x\(\right\))

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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\(\right\))=\(\frac{x+1}{x^3-4x^2+4x}\)

Textbook Question

Determine the following limits.

lim x→∞ (3 tan-1 x + 2)

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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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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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