Problem 27 - Single Variable Calculus

Question

Let $$ f(x)=\left\\{\begin{array}{ll} x+2 & \text { if } x \leq 1 \\ k x^{2} & \text { if } x>1 \end{array}\right. $$ Find the value of \(k\) that will make \(f\) continuous on \((-\infty, \infty)\).

Step-by-Step Solution

Answer

The value of k that makes f(x) continuous on \((-\infty, \infty)\) is 3.

Step 1: Find the left-side limit of f(x) as x approaches 1

Since x + 2 is defined for x ≤ 1, we will find the limit from the left as x approaches 1: \(\lim_{x\to1^-} f(x) = \lim_{x\to1^-} (x+2)\) As x approaches 1 from the left side, we can substitute 1 in the expression: \(\lim_{x\to1^-} f(x) = 1 + 2 = 3\)

Step 2: Find the right-side limit of f(x) as x approaches 1

Since kx^2 is defined for x > 1, we will find the limit from the right as x approaches 1: \(\lim_{x\to1^+} f(x) = \lim_{x\to1^+} (kx^2)\) As x approaches 1 from the right side, we can substitute 1 in the expression: \(\lim_{x\to1^+} f(x) = k(1)^2 = k\)

Step 3: Set the left-side limit equal to the right-side limit to find k

For f(x) to be continuous at x = 1, the left-side limit must equal the right-side limit. Therefore, we set 3 (left-side limit) equal to k (right-side limit): \(3 = k\)

Step 4: Identify the value of k that makes f(x) continuous

The equation 3 = k gives us the value of k that makes f(x) continuous on \((-\infty, \infty)\). Thus: \(k = 3\) The value of k that makes f(x) continuous on \((-\infty, \infty)\) is 3.