Theorem Interchanging the function & the Limit Sign Let : 1. f be a real function & is in the domain of f 2.The limit of is L 3.f is continues at L Then:

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Presentation transcript:

Theorem Interchanging the function & the Limit Sign Let : 1. f be a real function & is in the domain of f 2.The limit of is L 3.f is continues at L Then: The limit of is equal to f( L)

Examples (1)

Solutions

Question

Theorem The squeeze (Sandwich/pinching) theorem Let, and be sequences such that S n ≤ v n ≤ t n ; n ≥ k where k is any natural number

Example (1)

Example (2)

Theorem ( L' Hospital's rule for sequences ) Let f be a real function and a sequence such that S n = f(n) ; n ε IN. ( that’s is the restriction of f to IN ) The limit at infinity of f is L Then: The limit of the sequence is L

Examples (1)

Solutions

Questions

Hints

Question 6

Hints

Question 7

Question 8