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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 31
Chapter 2, Problem 31

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→2 (5x−6)^3/2

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1
Identify the type of limit problem: This is a direct substitution problem where we need to find the limit of a function as x approaches a specific value.
Recognize the function: The function given is \((5x - 6)^{3/2}\).
Apply direct substitution: Substitute \(x = 2\) into the function to evaluate the limit.
Calculate the expression inside the function: Compute \(5(2) - 6\) to simplify the expression inside the power.
Evaluate the power: Once the expression inside the function is simplified, raise it to the power of \(\frac{3}{2}\) to find the limit.

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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 continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 2.
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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. For the limit to exist, the function must not have any breaks, jumps, or asymptotes at the point of interest. The expression (5x−6) is a polynomial, which is continuous everywhere, including at x = 2.
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Intro to Continuity

Evaluating Limits of Composite Functions

When evaluating limits of composite functions, such as (5x−6) raised to the power of 3/2, it is essential to first find the limit of the inner function before applying the outer function. This process often involves substituting the limit value into the inner function and then applying the outer function to that result, ensuring that the operations are valid within the domain of the functions involved.
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Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question

Determine the following limits. 


lim x→∞ 6x2/(4x^2+√(16x4 + x2))

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→1 x^2 − 1 / x − 1

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The value of limx→3x2−9x−3{\(\displaystyle\]\lim\)_{x\(\to\)3}}\(\frac{x^2-9}{x-3}\) does not exist.

Textbook Question

Determine the following limits.

lim x→3- x − 4 / x^2 − 3x

Textbook Question

Determine the following limits. 


lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6

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Textbook Question

Determine the following limits.


b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2