Question

hw 5 - continuity section 1.4: problem 9 (1 point) find the value of the constant b that makes the following function continuous on (-∞, ∞). $f(x)=\begin{cases}7x - 1 & \text{if }xleq2\\- 6x + b & \text{if }x>2end{cases}$

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $x = a$, $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)$. Here $a = 2$.

Step2: Calculate left - hand limit

$\lim_{x
ightarrow 2^{-}}f(x)=\lim_{x
ightarrow 2}(7x - 1)$. Substitute $x = 2$ into $7x-1$, we get $7\times2-1=14 - 1=13$.

Step3: Calculate right - hand limit

$\lim_{x
ightarrow 2^{+}}f(x)=\lim_{x
ightarrow 2}(-6x + b)$. Substitute $x = 2$ into $-6x + b$, we get $-6\times2 + b=-12 + b$.

Step4: Set left - hand and right - hand limits equal

Since the function is continuous at $x = 2$, we set $-12 + b=13$.

Step5: Solve for $b$

Add 12 to both sides of the equation $-12 + b=13$. So $b=13 + 12=25$.

Answer:

$25$