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AP Calculus AB/BC73Updated Sep 8, 20262 pages

Understanding Limits with Trig and Pythagorean Identities

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Calculating limits involving trigonometric functions is a fundamental skill in calculus. When evaluating the limit of sin(x)/x as x approaches a value, we can apply strategic algebraic manipulations and trigonometric identities to find the solution.
Trig limit using Pythagorean identity – page 1

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Evaluating Limits with Trigonometric Functions

Ever wondered how to tackle those tricky limits with sine functions? The limit of sinxx/x is a classic problem that appears frequently in calculus exams.

To evaluate lim (x→a) sin(x)/xsin(x)/x, we can use the Pythagorean identity sin²xx + cos²xx = 1. This identity gives us a powerful tool to transform the expression into something more manageable.

The strategy involves multiplying both numerator and denominator by the same expression to create a useful pattern. In this case, we multiply by cosxx/cosxx, which equals 1 and doesn't change the value. This gives us:

lim (x→a) sin(x)/xsin(x)/x * [cosxx/cosxx] = lim (x→a) [(sinxx * cosxx)/x∗cos(x)x * cos(x)]

💡 When stuck on a limit problem, look for opportunities to multiply by a strategic form of 1 (like cosxx/cosxx) that can help simplify the expression!

Trig limit using Pythagorean identity – page 2

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Completing the Evaluation

Now that we've transformed our expression, we can continue simplifying to find the solution.

After multiplying by cosxx/cosxx, we have the expression [(sinxx * cosxx)/x∗cos(x)x * cos(x)]. Notice how cosxx appears in both the numerator and denominator? We can cancel it out, bringing us back to sinxx/x.

The key insight is that we now know lim (x→a) sin(x)/xsin(x)/x = 1. This is a fundamental result in calculus that you'll use repeatedly in future problems.

This limit is so important that it's worth memorizing: lim (x→0) sin(x)/xsin(x)/x = 1. While our example approached an arbitrary value 'a', the most common application is when x approaches zero.

🔑 Remember that canceling terms in limits requires careful attention - you can only cancel when the limit exists and isn't zero in the denominator!

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