To determine

To find: limx→∞⁡e3x-e-3xe3x+e-3x

Answer

Answer: 1

Explanation

 Limx→∞⁡e3x-e-3xe3x+e-3x=limx→∞⁡[e3xe3x+e-3x –e-3xe3x+e-3x]

= limx→∞⁡[e3xe3x+e-3x]- limx→∞e-3xe3x+e-3x…………………….. (A)

Divide first expression by e3x and second expression by e-3x in (A), we get

limx→∞⁡[11+e-6x]- limx→∞1e6x+1=11+limx→∞ e-6x- 1limx→∞ e6x+1=11+0-1∞=1.

We have used the fact that

i) For first expression of (A)  t=-6x ∞, as x∞ and limx→∞ e-6x=limt→-∞ et=0.

ii) For second expression of (A) t=6x ∞, as x∞ and limx→∞ e6x=limt→∞ et=∞.

Conclusion: 1