A function is defined by .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

A function is defined by h(x)=3x5+x2h(x)=\sqrt[5]{3x}+x^2. (a) Show that hh is continuous at x=0x=0. (b) Compute h′(x)h'(x) for x≠0x\neq 0. (c) Determine lim⁡x→0h′(x)\displaystyle \lim_{x\to 0}h'(x). Provide a complete limit calculation, including any algebraic simplifications or factorizations used. (d) Explain which term in h(x)h(x) governs the behavior of the slope near x=0x=0 and justify why the other term is negligible in this context. (e) Conclude whether hh possesses a vertical tangent at x=0x=0, supporting your claim with findings from parts (a)–(d).

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