Domanda del libro di testo
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 sin(1 − cos t) / (1 − cos t)
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 2, Problema 2.42
Limits and Infinity
Find the limits in Exercises 37–46.
x⁴ + x³
lim -----------------
x→∞ 12x³ + 128
Guida verificata passo dopo passo1
Identify the highest power of x in both the numerator and the denominator. In this case, the highest power in the numerator is x⁴ and in the denominator is x³.
Divide every term in the numerator and the denominator by x³, the highest power of x in the denominator.
Rewrite the expression: \( \lim_{{x \to \infty}} \frac{{x^4/x^3 + x^3/x^3}}{{12x^3/x^3 + 128/x^3}} \). This simplifies to \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12 + \frac{128}{x^3}}} \).
As x approaches infinity, the term \( \frac{128}{x^3} \) approaches 0 because the denominator grows much faster than the numerator.
Evaluate the limit: \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12}} \). As x approaches infinity, the dominant term is x, so the limit simplifies to \( \frac{x}{12} \), which approaches infinity.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the end behavior of rational functions.
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One-Sided Limits
Rational Functions
A rational function is a ratio of two polynomials. In the given limit problem, the numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 3. The degrees of the polynomials play a crucial role in determining the limit as x approaches infinity.
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Dominant Terms
In limits involving polynomials, the dominant term is the term with the highest degree, as it has the most significant impact on the function's value as x approaches infinity. For the given expression, the dominant term in the numerator is x⁴, and in the denominator, it is 12x³, which will dictate the limit's value.
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2:02Simplifying Trig Expressions Example 1
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Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
lim (4g(x))¹/³ = 2
x →0
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Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
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At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
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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) = x², x = 1
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Domanda del libro di testo
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)
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