I failed this task at my univiersity and i do not understand why. No feedback was given. I have to prove that cosh(x) and sinh(x) are continious. I proved it for cosh(x) and said the same principles could be applied to sinh(x). Here is my argument:
$cosh(x) = \frac{e^x + e^{-x}}{2}$
$e^x$ is continuous.
$e^{-x} = \frac{1}{e^x}$u and even though 1 is being divided by $e^x$ it is still continuous since having it divided by one does not change continuity.
When you add 2 continuous functions you get another continuous function, so:
$e^x + e^{-x}$ is still continuous.
Diving by 2 does not change continuity. $\frac{e^x + e^{-x}}{2}$ is therefore continuous. The same principle can be applied to sinh(x).
Is there something wrong with my argumentation or am I not explicit enough? What am I doing wrong?
By the way, differential calculus is not allowed in the task.
Best Answer
Your reasoning looks good, except when it comes to $e^{-x}$. True, that dividing $1$ by $e^{x}$ is still continuous, but why? The reason is that $e^{x}\neq 0$ for all $x\in\mathbb{R}$, and hence $e^{-x}=\frac{1}{e^{x}}$ is continuous as well since $e^{x}$ is.