Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.28

Capitolo 2, Problema 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))

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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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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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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

Pratica correlata

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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)

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Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

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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))