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

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

All textbooks

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 $1$ .

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.

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.

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.

Recommended video:

05:50

One-Sided Limits

Related Practice

Textbook Question

Determine the following limits.

lim x→∞ sin x / e^x

470

views

Textbook Question

If a function f represents a system that varies in time, the existence of lim ${$\displaystyle\[\lim$_{t$\rightarrow\]\infty$}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

372

views

Textbook Question

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$ , determine ${$\displaystyle\]\lim$_{x$\to$ a^{+}}}f$\left$(x$\right$)$ , ${$\displaystyle\]\lim$_{x$\to$ a^{-}}}f$\left$(x$\right$)$ , and ${$\displaystyle\]\lim$_{x$\to$ a}}f$\left$(x$\right$)$ .