Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

Verified step by step guidance

1. Recognize that the limit involves the trigonometric function tan(x) as x approaches 0.
2. Recall the identity for tangent: tan(x) = sin(x) / cos(x). This will help in simplifying the expression.
3. Consider the limit: lim x→0 tan(x) = lim x→0 (sin(x) / cos(x)).
4. Use the fact that as x approaches 0, sin(x) approaches 0 and cos(x) approaches 1. This simplifies the expression.
5. Apply the limit: lim x→0 (sin(x) / cos(x)) = (lim x→0 sin(x)) / (lim x→0 cos(x)) = 0 / 1.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for defining derivatives and integrals.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions that play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their properties and behaviors is key to solving limit problems involving these functions.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions.