Let 1\end{cases}"\u003eg(x)={x2+xif x1g\left(x\right)=\begin{cases}$$x^2+x$$ & \text{if }x1\end{cases}g(x)=⎩⎨⎧x2+xa3x+5if x1a. Determine the value of a for which gg is continuous from the left at 11.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.6.87

Chapter 2, Problem 2.6.87

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 $1$ .

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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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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>1a. Determine the value of a for which gg is continuous from the left at 11.

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Key Concepts

Continuity of Functions

Piecewise Functions

Limits

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 $1$ .