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 x1b. Determine the value of aa for which gg is continuous from the right at 11.

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 \u003e 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'.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.6.87b

Capitolo 2, Problema 2.6.87b

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 $1$ .

Guida verificata passo dopo passo

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'.

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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.

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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.

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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.

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