How to Find the Limit of a Trig Function
A limit, in mathematics, is the value a function approaches as a variable within the function approaches some number. Trigonometric functions are functions of angles and relate angles of triangles to the lengths of the sides of a triangle. The main strategy in calculating trig limits is to convert the trig functions into terms of sine and cosine using special relationships between trig functions. Some of the most important relationships include tan(x) = (sin(x) / cos(x)), cot(x) = (cos(x) / sin(x)), csc(x) = (1 / sin(x)), sec(x) = (1 / cos(x)), sin(2x) = 2sin(x)cos(x), cos(2x) = cos²(x) - sin²(x), sin²(x) = ( - cos(2x) / 2), cos²(x) = (1 + cos(2x) / 2), tan²(x) = (1 - cos(2x) / 1 + cos(2x)), sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x) and 1 + cot²(x) = csc²(x).
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Instructions
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Trigonometric Functions Without Exponents
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Check the problem to see if it contains any of the special trig limit properties. If so, apply them according to their rules.
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2
Convert any trig function into terms of sine and cosine, if possible. For example, to solve (limit as x ---> 0, from the right) cot(x), use the trig identity cot(x) = cos(x) / sin(x) and substitute the expression into the limit to put it into terms of sine and cosine. So, (limit as x ---> 0, from the right) cot(x) = (limit as x ---> 0, from the right) [cos(x) / sin(x)].
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3
Take the limit of the expression. For example, (limit as x ---> 0, from the right) [cos(x) / sin(x)] = cos(0) / sin(0) = 0 / 1 = 0.
Trigonometric Functions With Exponents
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4
Check the problem to see if it contains any of the special trig limit properties. If so, apply them according to their rules.
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5
Convert any trig function into terms of sine and cosine, if possible. For example, to solve (limit as x ---> 0, from the right) cot(x), use the trig identity tan^2(x) = sin^2(x) / cos^2(x) and substitute the expression into the limit to put it into terms of sine and cosine. So, (limit as x ---> 0) tan^2(x) = (limit as x ---> 0) [sin^2(x) / cos^2(x)].
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6
Factor the expression as much as possible. For example, (limit as x ---> 0) [1 - cos^3(x) / sin^2(x)] = (limit as x ---> 0) [(1 - cos(x))(1 + cos(x) + cos^2(x) / (1 + cos(x))(1 - cos(x)).
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7
Simplify the limit. For example, (limit as x ---> 0) [(1 - cos(x))(1 + cos(x) + cos^2(x) / (1 + cos(x))(1 - cos(x)) = (limit as x ---> 0) [1 + cos(x) + cos^2(x) / 1 + cos(x)].
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8
Take the limit of the expression. For example, (limit as x ---> 0) [1 + cos(x) + cos^2(x) / 1 + cos(x)] = (1 + cos(0) + cos^2(0) / 1 + cos(0)) = 1 + 1 + 1 / 1 + 1 = (3 / 2).
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