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 a. Determine the value of a for which g g is continuous from the left 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 left of X equals 2. The function is H X equals 2 X quad 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 answer is 1, B-1, C2, and D-2. Now what are we going to going to figure out which value of A makes the function continuous from the left of X equals 2? Well, to ensure that H of X is continuous from the left that X equals 2, the limit of H of X as X approaches 2 from the left must be equal to H of 2. In other words, The limit, OK, like we said, as X approaches 2 from the left of H of X must be equal to H of 2. So to figure out what the value A is, we have to evaluate the limit of the piece of the function that applies for X being less than 2 as X approaches 2 from the left and set it equal to A, the value of H of X at X equals 2. Now, when we look at our function here, if X is less than 2, then our function is 2 X squared minus 3X. So this is where we're going to substitute the value of 2 into our limit. So in other words, OK, from the left. OK. Then The limit as X approaches to. OK. H X is going to be equal to the limit as x approaches 2 from the left of 2 X2 minus 3 X. The final limit we can substitute 2 into our expression. So that's going to be 2 multiplied by 2 squad minus 3 multiplied by 2. 2 squared is 4 and 2 multiplied by 4 equals 8 by 3 multiplied by 2 equals 6 and 8 minus 6 equals 2. Thus, for H of X to be continuous from the left at X equals 2, the value of a must be equal to or limit here, which is 2. So that means then. OK, that A equals 2. If we look back in our answer choices, C is the correct answer. Thanks for watching everyone. I hope this video helped.

1. Limits and Continuity

Continuity

Calculus

1. Limits and Continuity

Continuity

2:53 minutes

Problem 2.6.87

Textbook Question

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

Verified step by step guidance

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

First, find the left-hand limit of g(x) as x approaches 1. Since g(x) = x^2 + x for x < 1, we calculate the limit of x^2 + x as x approaches 1 from the left.

Evaluate the expression x^2 + x at x = 1. This gives us the left-hand limit: lim_{x \to 1^-} (x^2 + x).

For g(x) to be continuous from the left at x = 1, this left-hand limit must equal g(1), which is 'a'. Therefore, set the left-hand limit equal to 'a'.

Solve the equation obtained in the previous step to find the value of 'a' that makes g(x) continuous from the left at x = 1.

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Key Concepts

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

Continuity of Functions

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 piecewise functions, this often involves ensuring that the left-hand limit and right-hand limit at the point of interest are equal to the function's value at that point.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this case, the function g(x) has three different definitions depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate these expressions is crucial for analyzing the function's behavior at specific points.

Limits

Limits describe the behavior of a function as it approaches a certain point. In the context of continuity, we need to evaluate the left-hand limit (as x approaches 1 from the left) and the right-hand limit (as x approaches 1 from the right) to determine if they are equal and match the function's value at that point, which is essential for ensuring continuity.

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