Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 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.

Verified step by step guidance

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.

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.

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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Textbook Question

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.

a. s(t)=−16t^2+80t+60 at t=3

Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

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)=x^1/3 <IMAGE>

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Textbook Question

Let $g$\left$(x$\right$)=$\frac{x^3-4x}{8\left|x-2\right|}$$ . <IMAGE>

Calculate $g$\left$(x$\right$)$ for each value of $x$ in the following table.

Textbook Question

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground?