AP Calculus AB · Interactive Practice
Advanced Trigonometric Limits
A fully mixed stream of randomized, multi-step trig limits. Problems may require identity rewrites, algebraic factoring, cancellation, standard trigonometric limits, or several of those ideas in combination. No derivatives or L'Hôpital's Rule are needed.
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Fully Mixed Advanced Practice
Find the limit.
Exact form preferred. Fractions and decimals are accepted; use √ to insert a square root, then enter its contents. Use → to move past the closing parenthesis. Typed sqrt(2) also works.
Required trig identity reference
Reciprocal / Quotient
\(\sin x=\frac1{\csc x},\quad \cos x=\frac1{\sec x},\quad \tan x=\frac1{\cot x}\)
\(\tan x=\frac{\sin x}{\cos x},\quad \cot x=\frac{\cos x}{\sin x}\)
Pythagorean
\(\sin^2x+\cos^2x=1\)
\(1+\tan^2x=\sec^2x,\quad 1+\cot^2x=\csc^2x\)
Double Angle / Power Reduction
\(\sin 2x=2\sin x\cos x\)
\(\cos 2x=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1\)
\(\sin^2x=\frac12(1-\cos 2x),\quad \cos^2x=\frac12(1+\cos 2x)\)
Even / Odd
\(\sin(-x)=-\sin x,\quad \cos(-x)=\cos x,\quad \tan(-x)=-\tan x\)
\(\csc(-x)=-\csc x,\quad \sec(-x)=\sec x,\quad \cot(-x)=-\cot x\)