Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 22

Capitolo 2, Problema 22

Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.

lim x →π cos² (x― tan x)

Guida verificata passo dopo passo

First, understand the expression for which you need to find the limit: \( \lim_{x \to \pi} \cos^2(x - \tan x) \). This involves the cosine function squared, evaluated at \( x - \tan x \).

Next, evaluate the behavior of \( x - \tan x \) as \( x \) approaches \( \pi \). Since \( \tan x \) is undefined at \( x = \pi \), consider the behavior of \( \tan x \) near \( \pi \).

Recognize that \( \tan x \) approaches 0 as \( x \) approaches \( \pi \) from either side, because \( \tan(\pi) = 0 \). Therefore, \( x - \tan x \) approaches \( \pi \) as \( x \to \pi \).

Substitute the limit of \( x - \tan x \) into the cosine function: \( \cos^2(x - \tan x) \to \cos^2(\pi) \). Recall that \( \cos(\pi) = -1 \), so \( \cos^2(\pi) = (-1)^2 = 1 \).

Conclude that the limit of the original expression is \( 1 \), as the squared cosine of \( \pi \) is \( 1 \). Thus, \( \lim_{x \to \pi} \cos^2(x - \tan x) = 1 \).

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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 the function's value at points where it may not be explicitly defined. For example, limits are essential for evaluating functions at points of discontinuity or for determining the behavior of functions at infinity.

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One-Sided Limits

Trigonometric Functions

Trigonometric functions, such as cosine and tangent, are periodic functions that relate angles to ratios of sides in right triangles. Understanding these functions is crucial for evaluating limits involving angles, especially when they approach specific values like π. The behavior of these functions near their critical points can significantly affect the limit's outcome.

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Percorso guidato

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Introduction to Trigonometric Functions

Continuity

Continuity refers to a property of functions where small changes in the input result in small changes in the output. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. This concept is vital when finding limits, as it helps determine whether a limit exists and if it can be evaluated directly.

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Intro to Continuity

Pratica correlata

Domanda del libro di testo

[Technology Exercise] Let f(t) = 1/t for t≠0.

a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.

b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.

c. What does your table indicate is the rate of change of f with respect to t at t=2?

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The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

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The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

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The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

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Domanda del libro di testo

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim h →0 ((x + h)² ― x²)/h

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Domanda del libro di testo

Finding Limits