Domanda del libro di testo
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→b (x − b)^50 − x + b / x − b
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 36
Determine the following limits.
lim x→−∞ (e^x cos x +3)
Guida verificata passo dopo passo1
Identify the behavior of each component of the function as \( x \to -\infty \).
Recognize that \( e^x \to 0 \) as \( x \to -\infty \) because the exponential function decays to zero.
Note that \( \cos x \) oscillates between -1 and 1 for all real numbers, including as \( x \to -\infty \).
Consider the product \( e^x \cos x \). Since \( e^x \to 0 \) and \( \cos x \) is bounded, \( e^x \cos x \to 0 \) as \( x \to -\infty \).
Combine the results: \( \lim_{x \to -\infty} (e^x \cos x + 3) = 0 + 3 = 3 \).
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Limits at Infinity
Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. Understanding how functions behave in these scenarios is crucial for determining their end behavior, which can often simplify complex expressions.
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05:50
One-Sided Limits
Exponential Functions
Exponential functions, such as e^x, grow rapidly as x increases and approach zero as x decreases towards negative infinity. This characteristic is essential for analyzing limits involving exponential terms, particularly when combined with oscillatory functions like cosine.
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6:13Exponential Functions
Trigonometric Functions
Trigonometric functions, like cos x, oscillate between -1 and 1 regardless of the value of x. When evaluating limits that include trigonometric functions, it's important to recognize their bounded nature, which can influence the overall limit when combined with other terms.
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Percorso guidato
6:04Introduction to Trigonometric Functions
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