Mastering Trigonometric Limits: Techniques for Evaluating Challenging Problems

Trig Limits

Can be solved with direct substitution (like normal functions)

Trigonometric limits can be challenging for some students. However, they can be evaluated using different approaches, such as algebraic manipulation, L’Hopital’s rule, and properties of limits. Here are some examples of how to evaluate trigonomic limits:

Example 1: Find the limit of sin(x)/x as x approaches 0.

Solution: We can use algebraic manipulation to evaluate this limit. We know that sin(0) equals 0, so the denominator x must also approach 0. Thus, we can rewrite the fraction as sin(x)/x = (sin(x)/x) * (x/x) = sin(x) / (x * 1) = sin(x). Therefore, the limit as x approaches 0 is sin(0) = 0.

Example 2: Evaluate the limit of tan(x)/x as x approaches 0.

Solution: We can use L’Hopital’s rule to evaluate this limit. First, we rewrite the fraction as sin(x)/cos(x) divided by x. Next, we take the derivative of both the numerator and the denominator. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Thus, the limit becomes (cos(x)/cos(x) – x*sin(x)/cos^2(x)) = 1 – (x*sin(x)/cos^2(x)). As x approaches 0, the second term in the equation goes to 0, and we are left with 1.

Example 3: Find the limit of (1-cos(x))/x^2 as x approaches 0.

Solution: We can use algebraic manipulation and L’Hopital’s rule to evaluate this limit. First, we can use the trigonometric identity sin^2(x)+cos^2(x)=1 to rewrite the numerator as 2sin^2(x)/2, which simplifies to sin^2(x). Thus, the limit becomes sin^2(x)/x^2. Next, we can rewrite this limit as (sin(x)/x)^2. This expression is in the form of a known limit, which is equal to 1. Therefore, the limit of (1-cos(x))/x^2 as x approaches 0 is 1.

In summary, trigonometric limits can be evaluated using algebraic manipulation, L’Hopital’s rule, or properties of limits. With enough practice and understanding of the techniques used, solving these types of problems can become easier for students.

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