Let 1\end{cases}"> g ( x ) = { x 2 + x if x < 1 a if x = 1 3 x + 5 if x > 1 g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases} g ( x ) = ⎩ ⎨ ⎧ x 2 + x a 3 x + 5 if x < 1 if x = 1 if x > 1 b. Determine the value of a a for which g g is continuous from the right at 1 1 .

Welcome back, everyone. In this problem, we want to figure out which value of A makes the following function continuous from the right of X equals 2. Our function is H X equals 2 X2 minus 3X if X is less than 2, A if X equals 2, and 4 X minus 1 if X is greater than 2. A says the value of A is 7, B says it's 9, C -7, and the D-9. Now how are we going to figure out which value of A is going to make the function continuous from the right of X equals 2? Well, to make the function h of X continuous from the right at X equals 2. Then the limit of H of X as X approaches 2 from the right, write that properly here. So its limit as HX approaches 2 from the right must be equal to the value of H of 2. So, in other words, if we can figure out the limit of our function as it approaches from the right, OK, then we can find a possible value of A. Now to figure it out as it approaches from the right, we have to ask ourselves, since H of X is a piecewise function, what part of the function applies when X is greater than 2? We'll notice here it would be 4 X minus 1. So that tells us that to find the limit of H of X as X approaches 2 from the right means that we're going to find the limit, as X approaches 2 from the right. OK, let me just write that properly here. OK. Of 4 X minus 1. Now, 4 X minus 1 is a linear expression. So to find that limit, we can simply substitute X equals 2 into the expression. So that is going to be equal to 4 multiplied by 2 minus 14 multiplied by 2 is 8, and 8 minus 1 equals 7. No, we can set the value of A equal to this limit, OK. So since the function must be continuous from the right at X equals 2, the value of a must be equal to the limit that we found, which is 7. Therefore, A is the correct answer. Thanks for watching everyone. I hope this video helped.

1. Limits and Continuity

Continuity

Calculus

1. Limits and Continuity

Continuity

2:49 minutes

Problem 2.6.87b

Textbook Question

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$
b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .

Verified step by step guidance

To determine the value of 'a' for which the function g(x) is continuous from the right at x = 1, we need to ensure that the right-hand limit of g(x) as x approaches 1 is equal to g(1).

The right-hand limit of g(x) as x approaches 1 is found by considering the expression for g(x) when x > 1, which is 3x + 5.

Calculate the right-hand limit: $ \lim_{{x \to 1^+}} g(x) = \lim_{{x \to 1^+}} (3x + 5) $.

Evaluate this limit by substituting x = 1 into the expression 3x + 5, which gives 3(1) + 5.

For g(x) to be continuous from the right at x = 1, set the right-hand limit equal to g(1), which is 'a'. Therefore, solve the equation 3(1) + 5 = a to find the value of 'a'.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function g(x) has three distinct cases depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate piecewise functions is crucial for determining their properties, such as continuity.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point from both sides equals the function's value at that point. For g(x) to be continuous at x=1, the limit as x approaches 1 from the left must equal the limit as x approaches 1 from the right, and both must equal g(1). This concept is essential for solving the problem.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, we need to find the left-hand limit (as x approaches 1 from values less than 1) and the right-hand limit (as x approaches 1 from values greater than 1) of g(x). Evaluating these limits will help determine the appropriate value of a that ensures continuity at x=1.

Watch next

Master Intro to Continuity with a bite sized video explanation from Patrick

Start learning

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}g(x)=⎩⎨⎧x2+xa3x+5if x<1if x=1if x>1b. Determine the value of aa for which gg is continuous from the right at 11.

Key Concepts

Piecewise Functions

Continuity

Limits

Watch next

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$
b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .