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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.6.88
2장, 문제 2.6.88

Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(9x² − x) − 3x)

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Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{9x^2 - x} - 3x) \).
Recognize that the expression involves a square root and a linear term, suggesting the use of the conjugate to simplify.
Multiply and divide the expression by the conjugate: \( \frac{(\sqrt{9x^2 - x} - 3x)(\sqrt{9x^2 - x} + 3x)}{\sqrt{9x^2 - x} + 3x} \).
Simplify the numerator using the difference of squares: \((\sqrt{9x^2 - x})^2 - (3x)^2 = 9x^2 - x - 9x^2 = -x\).
Rewrite the expression as \( \frac{-x}{\sqrt{9x^2 - x} + 3x} \) and analyze the behavior of the numerator and denominator as \( x \to \infty \) to find the limit.

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Limits at Infinity

Limits at infinity involve finding the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave asymptotically, often simplifying expressions to identify dominant terms that dictate the limit. In this problem, we analyze the limit as x approaches infinity to determine the function's end behavior.
추천 영상:
03:07
Cases Where Limits Do Not Exist

Conjugate Method

The conjugate method is a technique used to simplify expressions, especially those involving square roots. By multiplying and dividing by the conjugate, we can eliminate radicals and simplify the expression, making it easier to evaluate limits. This method is particularly useful in rationalizing differences involving square roots, as seen in the given problem.
추천 영상:
06:30
Disk Method Using y-Axis

Dominant Terms

Dominant terms are the parts of an expression that have the most significant impact on its value as the variable approaches infinity. Identifying these terms helps simplify the limit calculation by focusing on the highest degree terms, which dictate the behavior of the function. In this problem, recognizing that 9x² is the dominant term in the square root expression is key to finding the limit.
추천 영상:
2:02
Simplifying Trig Expressions Example 1
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교과서 질문

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) = 1/x, x = -2

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교과서 질문

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx → √3 1/x² = 1/3

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교과서 질문

Limits and Infinity


Find the limits in Exercises 37–46.


2x² + 3

lim -------------

x→⁻∞ 5x² + 7

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교과서 질문

If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.

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교과서 질문

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→0 x² sin (1/x) = 0


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교과서 질문

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ cot 2θ

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