17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))

Verified step by step guidance

1. Identify the form of the limit as x approaches π/2 from the left. Both the numerator tan(x) and the denominator 3/(2x - π) approach infinity, creating an indeterminate form of type ∞/∞.
2. Since the limit is in the indeterminate form ∞/∞, apply l'Hôpital's Rule, which allows us to differentiate the numerator and the denominator separately.
3. Differentiate the numerator: The derivative of tan(x) with respect to x is sec²(x).
4. Differentiate the denominator: The derivative of 3/(2x - π) with respect to x is -6/(2x - π)².
5. Re-evaluate the limit using the derivatives: lim_{x→π/2⁻} (sec²(x)) / (-6/(2x - π)²). Simplify the expression and evaluate the limit as x approaches π/2 from the left.

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 infinity. Evaluating limits is crucial for defining derivatives and integrals, which are core components of calculus.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating 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 separately. This rule simplifies the process of finding limits in complex functions.

Tangent Function

The tangent function, denoted as tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has vertical asymptotes where the cosine function is zero, which occurs at odd multiples of π/2. Understanding the behavior of the tangent function near these points is essential for evaluating limits involving tan(x).