Domanda del libro di testo
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x → -7 f(x)
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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 47
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim h→0 √16 + h − 4 / h
Guida verificata passo dopo passo1
Recognize that the limit \( \lim_{h \to 0} \frac{\sqrt{16 + h} - 4}{h} \) is an indeterminate form of type \( \frac{0}{0} \).
To resolve the indeterminate form, multiply the numerator and the denominator by the conjugate of the numerator: \( \frac{\sqrt{16 + h} - 4}{h} \times \frac{\sqrt{16 + h} + 4}{\sqrt{16 + h} + 4} \).
Simplify the expression: the numerator becomes \((\sqrt{16 + h} - 4)(\sqrt{16 + h} + 4) = (16 + h) - 16 = h\).
The expression simplifies to \( \frac{h}{h(\sqrt{16 + h} + 4)} \), and the \( h \) terms cancel out, leaving \( \frac{1}{\sqrt{16 + h} + 4} \).
Evaluate the limit by substituting \( h = 0 \) into the simplified expression: \( \lim_{h \to 0} \frac{1}{\sqrt{16 + h} + 4} \).
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Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as h approaches 0.
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05:50
One-Sided Limits
Rationalizing the Numerator
Rationalizing the numerator is a technique used to simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square root, making it easier to evaluate the limit. This method is particularly useful when dealing with limits that result in indeterminate forms.
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6:47Finding Limits Numerically and Graphically
Indeterminate Forms
Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is essential, as they often require additional techniques, such as factoring, rationalizing, or applying L'Hôpital's Rule, to resolve and find the actual limit.
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3:56Slope-Intercept Form
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