Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x → -7 f(x)
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x→0 f(x)
Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 1/x−1/4 / x − 4
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Textbook Question
Evaluate each limit.
lim x→−1 (x^2−4+ 3√x^2−9)
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Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(t)=(t^2−1)^3/2
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