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 2.7a

Chapter 2, Problem 2.7a

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.

Verified step by step guidance

Step 1: Understand the problem. We need to estimate the limit of the function $ \frac{\cos 2x}{\cos x - \sin x} $ as $ x $ approaches $ \frac{\pi}{4} $.

Step 2: Create a table of values for $ x $ approaching $ \frac{\pi}{4} $ from both the left and the right. Choose values such as $ \frac{\pi}{4} - 0.1 $, $ \frac{\pi}{4} - 0.01 $, $ \frac{\pi}{4} + 0.01 $, and $ \frac{\pi}{4} + 0.1 $.

Step 3: For each chosen value of $ x $, calculate $ \cos 2x $ and $ \cos x - \sin x $.

Step 4: Compute the value of the function $ \frac{\cos 2x}{\cos x - \sin x} $ for each $ x $ value in the table.

Step 5: Observe the trend of the function values as $ x $ approaches $ \frac{\pi}{4} $ and estimate the limit by rounding to four decimal places.

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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 limit of the function cos(2x) / (cos(x) - sin(x)) as x approaches π/4. Understanding limits is crucial for evaluating the function's value at points where it may not be directly computable.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are periodic functions that relate angles to ratios of sides in right triangles. In this problem, we are specifically dealing with cos(2x), cos(x), and sin(x). Familiarity with these functions and their properties, including their values at specific angles, is essential for estimating the limit accurately.

Table of Values

Creating a table of values involves calculating the function's output for various inputs approaching a specific point, in this case, π/4. This method provides a visual representation of how the function behaves near the limit and helps in estimating the limit by observing the trend of the values. It is a practical approach to understanding limits when direct substitution may lead to indeterminate forms.

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Related Practice

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Complete the following steps for the given functions.

a. Find the slant asymptote of $f$ .

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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

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.

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?