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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 11
Chapter 2, Problem 11

Determine the following limits.
lim x→1 √5x+6

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1
Identify the type of limit: This is a direct substitution limit problem where you need to evaluate the limit of a function as x approaches a specific value.
Substitute the value x = 1 directly into the function: \( \lim_{x \to 1} \sqrt{5x + 6} \).
Calculate the expression inside the square root: Substitute x = 1 into the expression 5x + 6 to get 5(1) + 6.
Simplify the expression: Calculate 5(1) + 6 to get the value inside the square root.
Evaluate the square root: Take the square root of the simplified expression 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 defining derivatives and integrals. In this case, we are interested in the limit of the function as x approaches 1.
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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. Continuous functions do not have breaks, jumps, or holes, making it easier to evaluate limits. The function √(5x + 6) is continuous at x = 1, allowing us to directly substitute the value into the function.
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Intro to Continuity

Substitution in Limits

Substitution is a technique used in evaluating limits where you replace the variable in the function with the value that the variable is approaching. If the function is continuous at that point, this method yields the limit directly. For the limit lim x→1 √(5x + 6), we can substitute x = 1 into the function to find the limit.
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Finding Limits by Direct Substitution
Related Practice
Textbook Question

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.


lim x→1 f(x) / g(x)−h(x)

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Textbook Question

Let g(t)=t−9t−3g\(\left\)(t\(\right\))=\(\frac{t-9}{\sqrt{t}\)-3}.

Make a conjecture about the value of limt→9t−9t−3{\(\displaystyle\]\lim\)_{t\(\to\)9}\(\frac{t-9}{\sqrt{t}\)-3}}.

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Textbook Question

Determine the following limits.

lim x→1000 18π^2

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Textbook Question

Given the function f(x)=−16x2+64xf\(\left\)(x\(\right\))=-16x^2+64x, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points (x,f(x))\(\left\)(x,f\(\left\)(x\(\right\))\(\right\)) and (2,f(2))\(\left\)(2,f\(\left\)(2\(\right\))\(\right\)), for x=1.5,1.9,1.99,1.999,x=1.5,1.9,1.99,1.999, and 1.99991.9999 (see figure).

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Textbook Question

Determine the following limits.

lim h→0 √5x + 5h − √5x / h, where x>0 Is constant

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Textbook Question

Given the function f(x)=−16x2+64xf\(\left\)(x\(\right\))=-16x^2+64x, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through (x,f(x))\(\left\)(x,f\(\left\)(x\(\right\))\(\right\)) and (2,f(2))\(\left\)(2,f\(\left\)(2\(\right\))\(\right\)) as xx approaches 22.

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