How do you prove that the triad of graphs of #y = cosh x , y = sinh x# and ( their bisector in the y-direction ) #y =(1/2) e^x# becomes asymptotic, as #x to oo#?

Question

How do you prove that the triad of graphs of #y = cosh x , y = sinh x# and ( their  bisector in the y-direction ) #y =(1/2) e^x#  becomes asymptotic, as #x to oo#?

William Abdalla · Accepted Answer

see below

I may have misunderstood the question but is it not a case of something like this for sinh and cosh: #lim_(x to oo) (e^x pm e^-x)/2# #=lim_(x to oo) 1/2e^x pm lim_(x to oo) 1/2 e^-x# #=lim_(x to oo) 1/2e^x pm 0# #=lim_(x to oo) 1/2e^x#

Charles Anctil · Answer

undefined To prove that the triad of graphs y = cosh x, y = sinh x, and y = (1/2) e^x becomes asymptotic as x approaches infinity, we can analyze their behavior at large values of x.

First, let's consider the graph of y = cosh x. The function cosh x grows exponentially as x increases. As x approaches infinity, the graph of cosh x also approaches infinity.

Next, let's examine the graph of y = sinh x. Similar to cosh x, sinh x also grows exponentially as x increases. As x approaches infinity, the graph of sinh x also approaches infinity.

Lastly, let's analyze the graph of y = (1/2) e^x, which represents the bisector in the y-direction. The function e^x grows exponentially faster than any polynomial function. As x approaches infinity, the graph of (1/2) e^x also approaches infinity.

Since all three graphs approach infinity as x approaches infinity, we can conclude that the triad of graphs y = cosh x, y = sinh x, and y = (1/2) e^x becomes asymptotic as x approaches infinity.