Evaluation of limits of the form 1∞

To evaluate the exponential limits of the form 1^∞, we use the following result.

Result: If   such that  exists, then,.

Proof:  Let

Remark: The above result can also be restated in the following form:

If   such that  exists, then

Particular cases:

Example: find the polynomial function f (x) of degree 6 satisfying:

Solution: Let f (x) = a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + a₅x⁵ + a₆x⁶.

Then,

⇒ a₀ = a₁ = a₂ = a₃ = 0, a₄ = 2

∴ f (x) = 2x⁴ + a₅x⁵ + a₆x⁶, where a₅, a₆ are real numbers.