Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.66
Chapter 3, Problem 3.5.66

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2

Verified step by step guidance
1
Recognize that the given limit can be interpreted as the derivative of a function at a specific point. The expression lim x→π/2 (cos x)/(x−π/2) resembles the definition of a derivative.
Recall the definition of the derivative of a function f at a point a: f'(a) = lim x→a (f(x) - f(a))/(x - a).
Identify the function f(x) = cos x and the point a = π/2. Notice that f(a) = cos(π/2) = 0.
Rewrite the limit in the form of the derivative definition: lim x→π/2 (cos x - cos(π/2))/(x - π/2). This matches the derivative definition for f(x) = cos x at x = π/2.
Conclude that the limit represents the derivative of f(x) = cos x at x = π/2, which is f'(π/2). To find this derivative, compute f'(x) = -sin x and evaluate it at x = π/2.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
Recommended video:
05:50
One-Sided Limits

Derivatives

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, recognizing the limit as a derivative allows for the application of derivative rules to evaluate the limit.
Recommended video:
05:44
Derivatives

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly useful in simplifying complex limit problems.
Recommended video:
Guided course
5:50
Power Rules
Related Practice
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

288
views
Textbook Question

27–76. Calculate the derivative of the following functions.

x2+93\(\sqrt\)[3]{x^2+9}

370
views
Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

187
views
Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

358
views
Textbook Question

Find d²/dx² (sin x + cos x).

269
views
Textbook Question

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

372
views