Question

In Exercises \(41-46,\) find (a) the right end behavior model, (b) the left end behavior model, and (c) any horizontal tangents for the function if they exist. $$y=\cot ^{-1} x$$

Short answer

(a) The right end behavior model is 0, (b) the left end behavior model is \(\pi\), and (c) the function does not have any horizontal tangents.

Step-by-step solution

End Behavior

To determine the right and left end behavior model, we consider the limits of the function as x approaches positive and negative infinity. End behavior of \(\cot^{-1} x\) is expressed as: (a) For Right end behavior, calculate: \(\lim_{{x \to \infty}} \cot^{-1} x\)(b) For left end behavior, calculate: \(\lim_{{x \to -\infty}} \cot^{-1} x\)

Solving the Limits

Calculate the limits (a) \(\lim_{{x \to \infty}} \cot^{-1} x\) = 0(b) \(\lim_{{x \to -\infty}} \cot^{-1} x\) = \(\pi\)as inverse cotangent function tends to 0 as x approaches infinity and \(\pi\) as x approaches negative infinity.

Determining Horizontal Tangents

To find any horizontal tangents, if they exist, we first find the derivative of the function, set it to zero and solve for x. The derivative of \(\cot^{-1} x\) is \(-\frac{1}{{1+x^2}}\). Setting this to zero, we find no real solutions, implying that there are no horizontal tangents for the \(\cot^{-1} x\) function.