Skip to main content
Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.6.93
Chapter 2, Problem 2.6.93

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → 0 (−1 / x²) = −∞

Verified step by step guidance
1
Understand the formal definition of a limit approaching negative infinity: For every positive number M, there exists a positive number δ such that for all x, 0 < |x| < δ implies f(x) < -M.
Identify the function f(x) = -1/x² and the limit statement lim x → 0 (−1 / x²) = −∞.
Given any positive number M, we need to find a δ > 0 such that for all x, 0 < |x| < δ, the inequality -1/x² < -M holds.
Rearrange the inequality -1/x² < -M to 1/x² > M, which implies x² < 1/M. Therefore, |x| < 1/√M.
Choose δ = 1/√M. Then, for all x such that 0 < |x| < δ, it follows that 1/x² > M, which satisfies the condition for the limit to be -∞.

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.

Limit Definition

The formal definition of a limit involves showing that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In this case, the limit is approaching negative infinity, which requires a modified approach to demonstrate that f(x) becomes arbitrarily large negative as x approaches 0.
Recommended video:
05:43
Definition of the Definite Integral

Behavior of Rational Functions

Understanding the behavior of rational functions, particularly as x approaches a value where the denominator tends to zero, is crucial. For the function −1/x², as x approaches 0, the denominator becomes very small, causing the function value to grow negatively without bound, leading to a limit of negative infinity.
Recommended video:
6:04
Intro to Rational Functions

Infinity in Limits

When dealing with limits that approach infinity, the concept of infinity in calculus is used to describe unbounded behavior. A limit approaching negative infinity means the function values decrease without bound as x approaches the specified point. This requires demonstrating that for any large negative number, the function can be made smaller than that number by choosing x sufficiently close to the limit point.
Recommended video:
03:07
Cases Where Limits Do Not Exist
Related Practice
Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

<IMAGE>

240
views
Textbook Question

Domains and Asymptotes


Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.


y = (√(x² + 4)) / x

288
views
Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0 (−1) / (x² (x + 1))

312
views
Textbook Question

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = −3/(x − 3)

309
views
Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=x³−3x²+4, P(2,0)

562
views
Textbook Question

Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.

499
views