Let

Question

Let $g\left(x ight)=\begin{cases}x^2+x & ext{if }x<1\ a & ext{if }x=1\ 3x+5 & ext{if }x>1\end{cases}$

b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .

Accepted Answer

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.

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 $1$ .

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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 $1$ .