Calculus I · Functions, limits, continuity, and asymptotic behavior
Rationalize a trigonometric radical difference
Problem
Evaluate the two-sided limit \[ \lim_{x\to0} \frac{\sqrt{1+\tan x}-\sqrt{1-\sin x}}{x}. \] First justify that both square roots are real on some open interval about zero. Then use a conjugate and standard radian trigonometric limits; show the algebra that determines the sign between $\tan x$ and $\sin x$.
Hint
Continuity makes both radicands positive sufficiently near zero.
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