Use definition of limit to prove <mo movablelimits="true" form="prefix">lim ( ln &#x2061;<!

Willow Pratt

Willow Pratt

Answered question

2022-06-29

Use definition of limit to prove lim ( ln ( n e + 1 n ) ) = 1

Answer & Explanation

Sariah Glover

Sariah Glover

Beginner2022-06-30Added 16 answers

Since ln is continuous
lim n ( ln ( n e + 1 n ) ) = ln ( lim n n e + 1 n ) = ln ( lim n ( e + 1 n ) ) = ln e = 1
you can also use ϵ δ definition since
| ln ( n e + 1 n ) 1 | = | ln ( n e + 1 n ) ln e | = | ln ( 1 + e n e n ) | = | ln ( 1 n e + 1 ) |
e x > x + 1 x > ln ( x + 1 )
x > ln ( x + 1 ) | ln ( 1 n e + 1 ) | < | 1 e n |
According to Archemedian Property there exist a natural number N such that
N e ϵ > 1
N e ϵ > 1 1 N e < ϵ
n > N | ln ( n e + 1 n ) 1 | < | 1 e n | < ϵ

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