 In this video, we provide the solution to question number nine from practice exam number three for math 1210 We're asked to evaluate the limit as x approaches infinity of hyperbolic secant of x There's kind of two ways you could approach this one is just by remembering the graph of hyperbolic secant Hyperbolic secant has this bump shape to it in particular as x approaches infinity We see that y will approach zero from above so that would indicate to us that the correct answer is in fact f So if you know the graph you can then use that to help us out here Another way of approaching that is just to rely on the definitions and identities of these hyperbolic functions So if we're trying to look for the limit of hyperbolic secant of x That's the same thing as taking the limit as x approaches infinity of 1 over cosh Right, which hyperbolic cosine we know its formula which we should know is gonna be 2 over e to the x plus e to the negative x As x approaches infinity here, so then if we do some arithmetic as x goes to infinity here We're gonna get 2 over e to the infinity Plus e to the negative infinity for which e to the infinity will be infinite e to the negative infinity will be zero Which case you get 2 over infinity which that tells us we get zero as well again We're approaching from above but that extra detail is not necessary here. We get the limit is going to be zero