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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 2.4.37
Capitolo 2, Problema 2.4.37

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

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Recognize that the limit involves the expression \( \frac{\sin \theta}{\sin 2\theta} \) as \( \theta \to 0 \). We need to manipulate this expression to use the known limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).
Rewrite \( \sin 2\theta \) using the double angle identity: \( \sin 2\theta = 2\sin \theta \cos \theta \). This gives us \( \frac{\sin \theta}{\sin 2\theta} = \frac{\sin \theta}{2\sin \theta \cos \theta} = \frac{1}{2\cos \theta} \).
Now, consider the limit \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} \). Since \( \cos \theta \to 1 \) as \( \theta \to 0 \), substitute this into the expression.
Evaluate the limit: \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} = \frac{1}{2 \cdot 1} \).
Conclude that the limit is \( \frac{1}{2} \).

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Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In calculus, limits are fundamental for understanding continuity, derivatives, and integrals. The notation limθ→0 indicates that we are examining the behavior of the function as θ approaches 0.
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Limits of Rational Functions: Denominator = 0

Sine Function Behavior

The sine function, sin(θ), is a periodic function that oscillates between -1 and 1. As θ approaches 0, sin(θ) behaves similarly to its argument, meaning sin(θ) approaches θ. This property is crucial for evaluating limits involving sine, particularly in the context of small angles.
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Graph of Sine and Cosine Function

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(θ)/g(θ) results in an indeterminate form, the limit can be evaluated by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when direct substitution does not yield a clear result.
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Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 3x - 4, x = 2

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Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 3x − 1 = 0

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Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → −∞ (2x + √(4x² + 3x − 2))

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Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

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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=√7−x, P(−2,3)

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Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

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