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 3.5.19
Chapter 3, Problem 3.5.19

Use Theorem 3.10 to evaluate the following limits.
lim x🠂2 (sin (x-2)) / (x2 - 4)

Verified step by step guidance
1
Theorem 3.10 refers to L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms like 0/0 or ∞/∞. First, check if the given limit is in an indeterminate form by substituting x = 2 into the expression.
Substitute x = 2 into the expression: (sin(x-2)) / (x^2 - 4). This results in (sin(0)) / (2^2 - 4), which simplifies to 0/0, confirming the indeterminate form.
Since the limit is in the 0/0 indeterminate form, apply L'Hôpital's Rule. This involves taking the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator: The derivative of sin(x-2) with respect to x is cos(x-2). Differentiate the denominator: The derivative of x^2 - 4 with respect to x is 2x.
Apply L'Hôpital's Rule by taking the limit of the new expression: lim x→2 (cos(x-2)) / (2x). Substitute x = 2 into this expression to evaluate the limit.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, evaluating the limit as x approaches 2 involves determining the behavior of the function (sin(x-2) / (x² - 4)) near that point, which may require simplification or application of limit laws.
Recommended video:
05:50
One-Sided Limits

Theorem 3.10 (L'Hôpital's Rule)

Theorem 3.10, commonly known as L'Hôpital's Rule, provides a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if such a form occurs, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator, and then re-evaluating the limit.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1

Trigonometric Limits

Trigonometric limits often involve functions like sine and cosine, which can exhibit unique behaviors near certain points. In this case, the limit of sin(x-2) as x approaches 2 is crucial, as it simplifies the evaluation of the overall limit. Understanding the properties of trigonometric functions helps in resolving limits involving these functions effectively.
Recommended video:
6:04
Introduction to Trigonometric Functions
Related Practice
Textbook Question

Find y'' for the following functions.

y = tan x

Textbook Question

Find y'' for the following functions.

y = x sin x

Textbook Question

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x(1−y²)+y³=0

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

g(x) = e^x / x²-1

Textbook Question

A challenging second derivative Find d²y/dx², where √y+xy=1.

Textbook Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = 12s³-8s²+12s/4s

2
views