Theorems · Theorem · complex analysis
Real.tendsto_one_add_pow_exp_of_tendsto
∀ {g : ℕ → ℝ} {t : ℝ},
Filter.Tendsto (fun n => ↑n * g n) Filter.atTop (nhds t) →
Filter.Tendsto (fun n => (1 + g n) ^ n) Filter.atTop (nhds (Real.exp t))The limit of (1 + g n) ^ n as (n : ℝ) → ∞ is exp t,
where t : ℝ is the limit of n * g n.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 278 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Real.expstatement · cited by 871
- Filter.Tendsto.compproof · cited by 560
- Real.rpow_natCastproof · cited by 93
- Filter.Tendsto.congrproof · cited by 70
- tendsto_natCast_atTop_atTopproof · cited by 51
- Nat.floor_natCastproof · cited by 19
- tendsto_smul_comp_nat_floor_of_tendsto_mulproof · cited by 4
- Real.tendsto_one_add_rpow_exp_of_tendstoproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.tendsto_choose_mul_pow_of_tendsto_mul_atTopproof · cited by 1