To determine the values of p and q for which the function f(x) is continuous for all x in R, we need to ensure that the function is continuous at every point, especially at the critical point x=0. For continuity at x=0, the left-hand limit (as x→0⁻), the right-hand limit (as x→0⁺), and the function value at x=0 must all be equal.

Step 1: Left-hand limit (x→0⁻)

For x < 0, f(x) = [sin((p+1)x) + sin(x)] / x

As x→0⁻, we can use the standard limit: lim_x→0 sin(ax)/x = a

So, lim_x→0⁻ f(x) = lim_x→0⁻ [sin((p+1)x)/x + sin(x)/x] = (p+1) + 1 = p + 2

Step 2: Right-hand limit (x→0⁺)

For x > 0, f(x) = [√(x + x²) - √x] / x^3/2

Simplify the expression:

√(x + x²) - √x = √x [√(1 + x) - 1]

So, f(x) = [√x (√(1+x) - 1)] / x^3/2 = (√(1+x) - 1) / x

Now, as x→0⁺, this is a 0/0 form. Apply rationalization:

(√(1+x) - 1)/x = [(√(1+x) - 1)(√(1+x) + 1)] / [x (√(1+x) + 1)] = (1+x - 1) / [x (√(1+x) + 1)] = x / [x (√(1+x) + 1)] = 1 / (√(1+x) + 1)

Therefore, lim_x→0⁺ f(x) = lim_x→0⁺ 1 / (√(1+x) + 1) = 1 / (1 + 1) = 1/2

Step 3: Function value at x=0

f(0) = q

Step 4: Equate for continuity at x=0

For continuity, left-hand limit = right-hand limit = f(0)

So, p + 2 = 1/2 = q

Thus, p + 2 = 1/2 ⇒ p = 1/2 - 2 = -3/2

And q = 1/2

Final Answer: p = -3/2, q = 1/2

Related Topics

Continuity of Functions: A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal. For piecewise functions, special attention is needed at the points where the definition changes.

Formulae

Standard Limit: lim_x→0 sin(ax)/x = a

Rationalization: Useful for evaluating limits involving square roots, e.g., (√(a) - √(b)) can be multiplied by (√(a) + √(b)) to simplify.