7) lim _(x arrow+infty) (cos x+1)/(cos x-1)

To find the limit of the given expression as approaches , we can use the fact that oscillates between -1 and 1. As approaches , does not approach a single value but instead oscillates indefinitely.To handle this, we can manipulate the expression to make it more amenable to limit evaluation. Let's rewrite the expression: First, we can multiply the numerator and the denominator by the conjugate of the denominator: This simplifies to: Next, we can use the Pythagorean identity to rewrite the denominator: Since is always positive and bounded between 0 and 1, the denominator is always negative and bounded between -1 and 0. As approaches , oscillates between -1 and 1, and the numerator oscillates between 0 and 4. However, the denominator approaches 0 from the negative side.Therefore, the fraction becomes increasingly large in magnitude but negative as approaches . Hence, the limit is: