Let f(x) = tan(x²) and g(x) = x|x| for |x|

Question

Let f(x) = tan(x²) and g(x) = x|x| for |x| < √(π/2). If p(x) = f(x)g(x), then which of the following statements is/are correct? I. p(x) is continuous at x = 0. II. p(x) is differentiable at x = 0. Select the answer using the code given below:

Accepted Answer

Both I and II. p(x) = tan(x²)·x|x|. Near x = 0, tan(x²) ≈ x², so p(x) ≈ x²·x|x| = x³|x|. This is continuous at x = 0 (value 0). For differentiability: p'(x) at 0 = lim[p(x)-p(0)]/x = lim x²|x|·tan(x²)/x² ... more carefully, p(x)/x ≈ x²|x| → 0 as x→0, so p'(0) = 0 exists. Hence both continuous and differentiable at x = 0.

Let f(x) = tan(x²) and g(x) = x|x| for |x| < √(π/2). If p(x) = f(x)g(x), then which of the following statements is/are correct? I. p(x) is continuous at x = 0. II. p(x) is differentiable at x = 0. Select the answer using the code given below:

Explanation

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