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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 2.4.27
Capitolo 2, Problema 2.4.27

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (tan 2x) / x

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Recognize that the limit involves the function tan(2x), which can be rewritten using the identity tan(2x) = sin(2x) / cos(2x). This gives us the expression lim(x→0) (sin(2x) / (x * cos(2x))).
To simplify the expression, focus on the term sin(2x) / x. Notice that this can be rewritten as (sin(2x) / 2x) * (2x / x).
Apply the limit property lim(θ→0) (sin(θ) / θ) = 1 to the term (sin(2x) / 2x). This implies that as x approaches 0, (sin(2x) / 2x) approaches 1.
Now, consider the term (2x / x), which simplifies to 2. Therefore, the expression (sin(2x) / x) simplifies to 2 * (sin(2x) / 2x).
Finally, evaluate the limit of the entire expression: lim(x→0) (2 * (sin(2x) / 2x) / cos(2x)). Since (sin(2x) / 2x) approaches 1 and cos(2x) approaches cos(0) = 1 as x approaches 0, the limit simplifies to 2 * 1 / 1.

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Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. In calculus, understanding limits is crucial for analyzing how functions behave near specific points, which is foundational for defining derivatives and integrals. The limit limθ→0 sin θ / θ = 1 is a standard result used to evaluate limits involving trigonometric functions.
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Limits of Rational Functions: Denominator = 0

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are essential tools for simplifying expressions and solving problems in calculus. For example, the identity tan x = sin x / cos x can be used to rewrite and simplify the expression tan 2x in terms of sine and cosine, aiding in the evaluation of limits.
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L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is particularly useful for evaluating limits involving complex expressions, such as limx→0 (tan 2x) / x.
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Power Rules
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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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Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



j. limx→2− f(x) = 2

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


y = cos (x) / 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))

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Limits of quotients


Find the limits in Exercises 23–42.


limx→−3 (2 − √(x² − 5)) / (x + 3)

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[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

         

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

         

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

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