The values of p and q for which the function f(x) = (p+1)x + x/x, & x 0 is continuous for all x in R , are: - Mathematics MCQ FIITJEE AIEEE 2011 (Set Q) 2026 | ExamTest.live

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The values of p and q for which the function f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x}, q, \frac{x + x ^{2} - x}{x ^{3/2}}, x < 0 x = 0 x > 0 is continuous for all x in R, are:

Solution & Step-by-step Explanation

For continuity at x = 0, LHL = RHL = f (0) = q .LHL: lim_{x \to 0^{-}} \frac{s i n ( p + 1 ) x + s i n x}{x} = lim_{x \to 0^{-}} (\frac{s i n ( p + 1 ) x}{( p + 1 ) x} (p + 1) + \frac{s i n x}{x}) = (p + 1) + 1 = p + 2 .RHL: lim_{x \to 0^{+}} \frac{x + x ^{2} - x}{x ^{3/2}} = lim_{x \to 0^{+}} \frac{x ( 1 + x - 1 )}{x x} = lim_{x \to 0^{+}} \frac{1 + x - 1}{x} .Rationalizing: lim_{x \to 0^{+}} \frac{1 + x - 1}{x ( 1 + x + 1 )} = lim_{x \to 0^{+}} \frac{1}{1 + x + 1} = 1/2 .Equating: q = 1/2 and p + 2 = 1/2 ⟹ p = - 3/2 .

Try it yourself before checking the explanation above.

The values of p and q for which the function f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x}, q, \frac{x + x ^{2} - x}{x ^{3/2}}, x < 0 x = 0 x > 0 is continuous for all x in R, are:

p = \frac{5}{2}, q = \frac{1}{2}

p = - \frac{3}{2}, q = \frac{1}{2}

p = \frac{1}{2}, q = \frac{3}{2}

p = \frac{1}{2}, q = - \frac{3}{2}

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