Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→0 √(7 + sec²x)

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the limit of the function \( \sqrt{7 + \sec^2 x} \) as \( x \) approaches 0.
2. Step 2: Recall the definition of the secant function. The secant function is defined as \( \sec x = \frac{1}{\cos x} \). Therefore, \( \sec^2 x = \frac{1}{\cos^2 x} \).
3. Step 3: Substitute \( \sec^2 x \) in the original function. The expression becomes \( \sqrt{7 + \frac{1}{\cos^2 x}} \).
4. Step 4: Evaluate the limit as \( x \to 0 \). As \( x \to 0 \), \( \cos x \to 1 \), so \( \cos^2 x \to 1 \). Therefore, \( \frac{1}{\cos^2 x} \to 1 \).
5. Step 5: Simplify the expression inside the square root. As \( x \to 0 \), the expression inside the square root becomes \( 7 + 1 = 8 \). Thus, the limit of the function is \( \sqrt{8} \).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as secant (sec), are periodic functions that relate angles to ratios of sides in right triangles. The secant function is defined as the reciprocal of the cosine function. Understanding how these functions behave, especially near specific angles like 0, is crucial for evaluating limits involving them.

Secant Function Behavior

The secant function, sec(x), approaches infinity as x approaches odd multiples of π/2, where the cosine function is zero. However, as x approaches 0, sec(x) approaches 1 since cos(0) = 1. This behavior is important for evaluating the limit of the expression √(7 + sec²x) as x approaches 0.