Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.
lim x→π/2^− tan x

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this case, we are interested in the limit of the tangent function as x approaches π/2 from the left. Understanding limits helps in analyzing the behavior of functions near points of discontinuity or asymptotes.

Tangent Function

The tangent function, defined as tan(x) = sin(x)/cos(x), is periodic and has vertical asymptotes where the cosine function equals zero, such as at x = π/2. This characteristic leads to the function approaching infinity as x approaches these points from the left or right, which is crucial for evaluating the limit in the question.

Graphing Functions

Graphing functions provides a visual representation of their behavior, including limits and asymptotes. By sketching the graph of y = tan(x) over the specified window, one can observe the function's approach to infinity as x nears π/2, confirming the analytical limit found. This visual check enhances understanding of the function's properties.