Domanda del libro di testo
Determine the following limits.
lim x→1 x^3 − 7x^2 + 12x / 4 − 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 2.6.53
Evaluate each limit.
Guida verificata passo dopo passo1
Identify the limit expression: \( \lim_{x \to 3} \sqrt{x^2 + 7} \).
Recognize that this is a direct substitution problem since the function \( \sqrt{x^2 + 7} \) is continuous at \( x = 3 \).
Substitute \( x = 3 \) directly into the expression: \( \sqrt{3^2 + 7} \).
Simplify the expression inside the square root: \( 3^2 + 7 = 9 + 7 \).
Calculate the square root of the simplified expression: \( \sqrt{16} \).
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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 continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.
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05:50
One-Sided Limits
Continuous Functions
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is essential for evaluating limits, as it allows us to directly substitute the value into the function if it is continuous. The function in the limit question, √(x² + 7), is continuous everywhere since it is a polynomial under a square root.
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05:34Intro to Continuity
Substitution in Limits
Substitution is a technique used in limit evaluation where you replace the variable in the function with the value it approaches. If the function is continuous at that point, this method yields the limit directly. For the given limit, substituting x = 3 into the function √(x² + 7) will provide the limit value without further manipulation.
Video consigliato:
05:21Finding Limits by Direct Substitution
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Determine the following limits.
lim x→∞ sin x / e^x
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Domanda del libro di testo
Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = x^2−25 / x−5; a=5
290
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Domanda del libro di testo
Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = {x^2+1 if x≤−1
3 if x>−1; a=−1
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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→3 x − 3 /|x − 3|
243
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Domanda del libro di testo
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(x)=(2x−3)^2/3
252
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