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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 2.2.55a
Chapter 2, Problem 2.2.55a

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

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To estimate the limits using a graphing utility, first understand the expressions: lim x→0 tan(2x) / sin(x), lim x→0 tan(3x) / sin(x), and lim x→0 tan(4x) / sin(x). These involve trigonometric functions where x approaches 0.
Graph each function separately: y = tan(2x) / sin(x), y = tan(3x) / sin(x), and y = tan(4x) / sin(x). Use a graphing calculator or software to plot these functions over an interval around x = 0, such as [-0.1, 0.1].
Observe the behavior of each graph as x approaches 0. The limit is the value that the function approaches as x gets closer to 0. Look for a horizontal line or a point where the function stabilizes near x = 0.
For each function, note the y-value that the graph approaches as x approaches 0. This y-value is the estimated limit for each expression.
Compare the estimated limits for each function. Notice any patterns or differences in the limits as the coefficient of x in the tangent function changes from 2 to 3 to 4.

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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. In this case, we are interested in the limits of the functions tan(kx) and sin(x) as x approaches 0. Understanding limits is crucial for evaluating expressions that may be indeterminate at specific points.
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Trigonometric Functions

Trigonometric functions, such as sine and tangent, are periodic functions that relate angles to ratios of sides in right triangles. The behavior of these functions near specific points, like x = 0, is essential for evaluating limits. For small angles, both tan(x) and sin(x) can be approximated, which simplifies the limit calculations.
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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can be particularly useful when dealing with limits involving trigonometric functions.
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Related Practice
Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^− h(x)

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Textbook Question

Determine the following limits.


a. limx→1+x−3x2−5x+4{\(\displaystyle\]\lim\)_{x\(\to\)1^{+}}\(\frac{x-3}{\sqrt{x^2-5x+4}\)}}

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.


f(x)=x1/3 <IMAGE>

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Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


a. lim x→−2^+ f(x)

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