Question

determine the end - behavior of the following transcendental function by evaluating appropriate limits. then provide a simple sketch of the associated graph showing asymptotes if they exist. f(x)=sin(x)
choose the correct end - behavior of the given function
a. lim_{x→∞}sin(x)= and lim_{x→ - ∞}sin(x)=
b. lim_{x→∞}sin(x) does not exist and lim_{x→ - ∞}sin(x) does not exist

Explanation:

Step1: Recall the nature of sine - function

The sine function \(y = \sin(x)\) is a periodic function with period \(T = 2\pi\). It oscillates between - 1 and 1 for all real - valued \(x\).

Step2: Evaluate the limit as \(x\to+\infty\)

As \(x\) approaches positive infinity (\(x\to+\infty\)), the values of \(\sin(x)\) keep oscillating between - 1 and 1. So, \(\lim_{x\to+\infty}\sin(x)\) does not exist.

Step3: Evaluate the limit as \(x\to-\infty\)

As \(x\) approaches negative infinity (\(x\to-\infty\)), the values of \(\sin(x)\) also keep oscillating between - 1 and 1. So, \(\lim_{x\to-\infty}\sin(x)\) does not exist.

Answer:

B. \(\lim_{x\to+\infty}\sin(x)\) does not exist and \(\lim_{x\to-\infty}\sin(x)\) does not exist