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Magazine Chapter 2: Problem 43
Exer. \(43-44:\) Show that \(f\) is continuous at the number \(a\). $$ f(x)=\sqrt{5 x+9} ; \quad a=8 $$
Short Answer
Expert verified
The function \( f(x) = \sqrt{5x+9} \) is continuous at \( x = 8 \).
Step by step solution
01
Understand the Definition of Continuity
A function \( f \) is continuous at a point \( a \) if \( \lim_{{x \to a}} f(x) = f(a) \). We need to check this condition for \( f(x) = \sqrt{5x+9} \) at \( a = 8 \).
02
Find \( f(a) \)
Calculate \( f(8) \) by substituting \( x = 8 \) into the function. \[ f(8) = \sqrt{5\cdot8 + 9} = \sqrt{40 + 9} = \sqrt{49} = 7 \]
03
Calculate the Limit \( \lim_{{x \to 8}} f(x) \)
Determine \( \lim_{{x \to 8}} \sqrt{5x+9} \). Since \( f(x) = \sqrt{5x+9} \) is a continuous function (composed of a polynomial inside a square root), you can directly substitute \( x = 8 \): \[ \lim_{{x \to 8}} \sqrt{5x+9} = \sqrt{5\cdot8 + 9} = \sqrt{49} = 7 \]
04
Compare \( \lim_{{x \to 8}} f(x) \) and \( f(8) \)
We have found that \( \lim_{{x \to 8}} f(x) = 7 \) and \( f(8) = 7 \). Since they are equal, \( f \) is continuous at \( x = 8 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit
In calculus, the concept of a limit is a fundamental building block for understanding changes and behavior in functions. A limit explains what value a function approaches as the input gets closer to a given point. For the function \( f(x) = \sqrt{5x+9} \) at \( a = 8 \), we need to find the value that \( f(x) \) approaches as \( x \) gets infinitely close to 8.
This involves calculating \( \lim_{{x \to 8}} \sqrt{5x+9} \). Some key points about limits include:
This involves calculating \( \lim_{{x \to 8}} \sqrt{5x+9} \). Some key points about limits include:
- The limit answers the question: "What does \( f(x) \) get close to as \( x \) approaches a particular value?"
- If you can substitute \( x = a \) directly into \( f(x) \) without changing the outcome, the limit is straightforward.
- In our case, the limit is crucial in proving that the function is continuous at a particular point.
Continuous Function
A continuous function means there are no gaps, jumps, or breaks in the function at any point in its domain. For \( f(x) = \sqrt{5x+9} \), we need to demonstrate that it's continuous at \( a = 8 \) by ensuring two conditions:
- \( \lim_{{x \to a}} f(x) \) exists.
- \( \lim_{{x \to a}} f(x) = f(a) \).
Calculus
Calculus is the branch of mathematics that studies change, using concepts like limits, derivatives, and integrals. In this exercise, calculus helps analyze whether the given function \( f(x) = \sqrt{5x+9} \) behaves smoothly as \( x \) approaches 8. This process involves:
- Applying the limit to establish continuity.
- Understanding how small changes in \( x \) influence \( f(x) \).
Polynomial
A polynomial is an expression consisting of variables and coefficients, combined using operations of addition, subtraction, and multiplication. In our context, the polynomial inside the square root is \( 5x + 9 \).
Key characteristics of polynomials:
Key characteristics of polynomials:
- They are smooth and continuous over the real numbers.
- The expression \( 5x + 9 \) is a linear polynomial.
Square Root
The square root function extracts the principal square root of a number, making it a unique and essential operation. Within \( f(x) = \sqrt{5x+9} \), the square root provides a curve derived from the underlying polynomial \( 5x + 9 \).
- Aside from non-negative values, square roots yield real, positive outputs.
- They must be accompanied by a non-negative argument to ensure real results.
