Evaluating Trigonometric Limits with Identities

Evaluating Limits Using Trigonometric Identities

Limits of trigonometric functions often require specific techniques to evaluate them accurately, especially when dealing with indeterminate forms or when the limit involves trigonometric expressions. In such cases, trigonometric identities become a valuable tool for simplifying expressions and finding the exact limit values.

Limits of Trigonometric Functions

To understand how trigonometric identities can assist in evaluating limits, let's first examine the behavior of trigonometric functions as the input variable approaches specific values.

1. Sine and cosine functions: As (x) approaches (0), ( \sin{x} ) oscillates between (-1) and (1) with decreasing amplitude, and ( \cos{x} ) oscillates between (1) and (-1) with decreasing amplitude.
2. Tangent function: As (x) approaches (0), ( \tan{x} ) becomes large and positive or large and negative, depending on whether (x) approaches (0) from the right or left, respectively.
3. Secant, cosecant, and cotangent functions: As (x) approaches (\frac{\pi}{2}), ( \sec{x} ) approaches (0) from the right, and ( \sec{x} ) approaches (-\infty) from the left. On the other hand, ( \csc{x} ) approaches (\infty) from the right and (-\infty) from the left, and ( \cot{x} ) approaches (0) from the left and (-\infty) from the right.

Evaluating Limits Using Trigonometric Identities

In evaluating limits involving trigonometric functions, we may use the following trigonometric identities:

1. Linear properties of limits: If (f(x)) and (g(x)) have limits (L_f) and (L_g) as (x) approaches (a), then (f(x) + g(x)), (f(x)g(x)), and (\frac{f(x)}{g(x)}) have limits (L_f + L_g), (L_f \cdot L_g), and (\frac{L_f}{L_g}), respectively, as (x) approaches (a).
2. Sum-to-product formula: (a \cos{x} + b \sin{x} = \sqrt{a^2 + b^2} \cdot \cos{(x - \theta)}), where (\cos{\theta} = \frac{a}{\sqrt{a^2 + b^2}}) and (\sin{\theta} = \frac{b}{\sqrt{a^2 + b^2}}).
3. Double-angle formula: (2 \sin^2{(x)} = 1 - \cos{(2x)}).
4. Substitution: Replace (x) with an expression that simplifies the limit.

Examples

1. Evaluate the limit as (x) approaches (0) of (\frac{\sin{x} - \cos{x}}{x}):

[ \lim_{x \to 0} \frac{\sin{x} - \cos{x}}{x} = \lim_{x \to 0} \frac{2 \sin{\frac{x}{2}} \cos{\frac{x}{2}} - \cos{\frac{x}{2}}}{x} \stackrel{\text{substitution}}{=} \lim_{x \to 0} \frac{2 \sin{\frac{x}{2}} \left(1 - \cos{\frac{x}{2}}\right)}{\frac{x}{2}} = \lim_{x \to 0} \frac{2 \sin{\frac{x}{2}} \sin{\frac{x}{2}}}{\frac{x}{2}} = \lim_{x \to 0} 2 \cdot 2 \frac{\sin{x}}{x} = 2 ]

2. Evaluate the limit as (x) approaches (\pi/2) of (\frac{\csc{x}}{\sec{x}}):

[ \lim_{x \to \frac{\pi}{2}} \frac{\csc{x}}{\sec{x}} = \lim_{x \to \frac{\pi}{2}} \frac{1}{\cos{x}} \cdot \frac{1}{\sin{x}} = \lim_{x \to \frac{\pi}{2}} \frac{1}{\cos{x} \sin{x}} = \lim_{x \to \frac{\pi}{2}} \frac{1}{- \sin^2{(x)}} = -\lim_{x \to \frac{\pi}{2}} \frac{1}{\sin^2{(x)}} = -\frac{1}{0} \text{ (not defined)} ]

However, we can factor out the (\sin{x}) to get:

[ \lim_{x \to \frac{\pi}{2}} \frac{1}{\cos{x} \sin{x}} = \lim_{x \to \frac{\pi}{2}} \frac{1}{\sin{(x - \frac{\pi}{2})}} = -\lim_{x \to 0} \frac{1}{\sin{x}} = - \frac{1}{0} \text{ (not defined)} ]

Instead, use the double-angle formula:

[ \lim_{x \to \frac{\pi}{2}} \frac{1}{\cos{x} \sin{x}} = \lim_{x \to \frac{\pi}{2}} \frac{1}{2 \sin^2{\frac{x}{2}}} = \lim_{x \to 0} \frac{1}{2 \sin^2{\frac{x}{2}}} = \frac{1}{2} ]

Conclusion

Mastering the use of trigonometric identities in evaluating limits is an essential skill for students and professionals in mathematics and its applications. These identities allow us to simplify complex expressions, use linear properties of limits, and rewrite trigonometric limits in terms of more basic limits. The ability to apply these techniques to evaluate difficult limits accurately is a valuable tool in problem-solving.