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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 2.2.47
Chapter 2, Problem 2.2.47

Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

Verified step by step guidance
1
Recognize that the limit involves the trigonometric function tan(x) as x approaches 0.
Recall the identity for tangent: tan(x) = sin(x) / cos(x). This will help in simplifying the expression.
Consider the limit: lim x→0 tan(x) = lim x→0 (sin(x) / cos(x)).
Use the fact that as x approaches 0, sin(x) approaches 0 and cos(x) approaches 1. This simplifies the expression.
Apply the limit: lim x→0 (sin(x) / cos(x)) = (lim x→0 sin(x)) / (lim x→0 cos(x)) = 0 / 1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for defining derivatives and integrals.
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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions that play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their properties and behaviors is key to solving limit problems involving these functions.
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Introduction to Trigonometric Functions

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions.
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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?

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Textbook Question

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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Textbook Question

Using the Sandwich Theorem


If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

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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=√7−x, P(−2,3)

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Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

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Textbook Question

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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