Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 12

Chapter 2, Problem 12

Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.

lim x→a (x² ― a²)/(x⁴ ― a⁴)

Verified step by step guidance

First, recognize that the expression (x² - a²)/(x⁴ - a⁴) is a rational function. To find the limit as x approaches a, we need to simplify the expression, especially since direct substitution would lead to an indeterminate form 0/0.

Notice that both the numerator and the denominator are differences of squares. The numerator x² - a² can be factored as (x - a)(x + a). Similarly, the denominator x⁴ - a⁴ can be factored using the difference of squares twice: first as (x² - a²)(x² + a²), and then further factor x² - a² as (x - a)(x + a).

After factoring, the expression becomes ((x - a)(x + a))/((x - a)(x + a)(x² + a²)). Cancel the common factors (x - a)(x + a) from the numerator and the denominator, provided x ≠ a.

The simplified expression is now 1/(x² + a²). This expression is no longer indeterminate as x approaches a.

Finally, evaluate the limit by substituting x = a into the simplified expression, which is now straightforward since it no longer results in an indeterminate form.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

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 specific points, which is crucial for analyzing continuity, derivatives, and integrals. In this case, we are interested in the limit of a rational function as x approaches a.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors, such as asymptotes and discontinuities, depending on the values of x. In the given limit problem, both the numerator and denominator are polynomials, and their degrees will influence the limit's existence and value as x approaches a.

Factoring and Simplifying

Factoring and simplifying expressions is a key technique in calculus for evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. By factoring the numerator and denominator, we can often cancel common terms, making it easier to find the limit as x approaches a. This process is essential for resolving the limit in the given exercise.

Recommended video:

6:36

Simplifying Trig Expressions

Related Practice

Textbook Question

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.

lim (x lim g(x)) = 2

x→-4 x→0

188

views

Textbook Question

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.

379

views

Textbook Question

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

290

views

Textbook Question

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.

lim ((4―g(x)) / x ) = 1

x→0

275

views

Textbook Question