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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 2.28
Chapter 2, Problem 2.28

Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.


5 ―x²
lim ------------- = 0
x → ―2 (√g(x))

Verified step by step guidance
1
Identify the function g(x) given in the problem. Here, g(x) is not explicitly defined, but we are given an expression involving g(x) in the limit.
Rewrite the limit expression in a more standard form. The problem states: lim (5 - x²) / (√g(x)) as x approaches -2. This suggests that g(x) is related to the expression in the denominator.
Assume that g(x) is such that the expression (5 - x²) / (√g(x)) is defined and continuous around x = -2. This might involve simplifying or rationalizing the expression.
Evaluate the limit by substituting x = -2 into the expression, if possible. If direct substitution leads to an indeterminate form, consider using algebraic manipulation or L'Hôpital's Rule to resolve the indeterminate form.
Conclude by stating the limit value, ensuring that the steps taken are consistent with the properties of limits and continuity.

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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. It helps in understanding how functions behave near specific points, even if they are not defined at those points. For example, the limit of a function can indicate whether it approaches a finite value, infinity, or does not exist as the input approaches a particular value.
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One-Sided Limits

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is crucial when evaluating limits, as discontinuities can lead to different limit behaviors. Understanding continuity helps in determining whether the limit can be directly substituted or requires further analysis.
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Intro to Continuity

Rational Functions

Rational functions are ratios of polynomials, and their limits can often be evaluated by simplifying the expression. When finding limits involving rational functions, it is important to identify any points of discontinuity, such as holes or vertical asymptotes, which can affect the limit's value. Techniques like factoring and canceling common terms are often employed to simplify the limit calculation.
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Intro to Rational Functions
Related Practice
Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Textbook Question

At what points are the functions in Exercises 13–30 continuous?


y = cos (x) / x

Textbook Question

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?


lim x → 0 tan (π/4 cos (sin x¹/³))

Textbook Question

If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


llimx→0 (x −x cos x) / sin² 3x

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?

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