Limits of Trig Functions

Trig Limits from Unit Circle

We evaluate simple trig limits such as using the coordinates of the unit circle.

Recall, that on a unit circle, the cosine and sine of the angle in standard position are represented by the x and y coordinates of the circumference point due to the definitions of sine and cosine and the fact that a unit circle has a radius = 1.

  a = 0 a = o/2 a = o a = 3o/2
1 0 -1 0
0 1 0 -1
0   0  


For the reciprocal functions, secant, cosecant and cotangent, flip the fractions. Recall that division by 0 is , and division by is 0.

Limits for other values of h such as o/3, o/6, and o/4 are found using the ratios of the sides in the 30-60-90 triangle () or the 45-45-90 () triangle.



Two Useful Limits

When our trig limit is not simple -- involves a fraction and multiple trig functions, we use 2 useful limit formats with trig identities to simplify the question.

1)A Limit of Sin h:

Note the argument of the sine function = the variable!

So, any expression which can be put into this format has a value = 1.


Notice how we multiplied top and bottom by the same value (17).

We only changed the format of the expression -- not the value.

Here's an exam classic.


Notice what we did. In order to set up the format , or its reciprocal,

we multiplied top and bottom by 24x, but placed the terms strategically.

Then we assign the value 1 to , or its reciprocal.

We must make sure that the effect of every term we introduce into the fraction is eliminated by something else. We put "3x" in the denominator of the 1st fraction, so we put 3 and x in the numerator further on. We did the same with "8x". So in the end, it all "cancels" out but we get a value for the limit.

One more example:


Here, we break the expression up into a product of fractions like before.

Notice the "4" in the denominator which comes from (2x) 2 = 4x 2

2) A Limit of Cos h:

The other useful limit "format" involves cos h as h approaches zero.

Note the argument of the cosine function = the variable!


1) Find

2) Find



Mixed Bag:

Here are a couple of examples on how to use trig identities to simplify the limit expressions.


(in the last fraction, cross out the sin x, set cos 0 = 1)



Find the limits. Hint: break up the fractions to fit the formats.

1) 2) 3)
4) 5) 6)


1) 2)
3) 4)
5) 6)


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