Question

let (f) be the function defined above, where (c) is a constant. if (f) is continuous at (x = 1), what is the value of (c)?
(f(x)=\begin{cases}5 + cx& \text{for }x < 1\\10 + 6ln x& \text{for }xgeq1end{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 $a = 1$.

Step2: Calculate left - hand limit

$\lim_{x
ightarrow1^{-}}f(x)=\lim_{x
ightarrow1^{-}}(5 + cx)$. Substituting $x = 1$ into $5+cx$, we get $5 + c(1)=5 + c$.

Step3: Calculate right - hand limit

$\lim_{x
ightarrow1^{+}}f(x)=\lim_{x
ightarrow1^{+}}(10 + 6\ln x)$. Substituting $x = 1$ into $10 + 6\ln x$, since $\ln1=0$, we have $10+6\ln1 = 10$.

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

Since $f(x)$ is continuous at $x = 1$, $5 + c=10$.

Step5: Solve for $c$

Subtract 5 from both sides of the equation $5 + c=10$, we get $c=10 - 5=5$.

Answer:

5