by @mes on Math Easy Solutions 
View my bio on Blurt.media: https://blurt.media/c/mes [![Exercise 3: Convergent Sequence inside a Continuous Function](https://blurt.media/lazy-static/previews/accf7d8a-20d9-46eb-8005-2fb263827989.jpg)](https://blurt.media/w/ssNnxSdxKJqy7z7uTkso2q) 
 
 In this video I prove Theorem 3, which states that a convergent sequence with limit L can be plugged directly into a continuous function at L, thus obtaining f(L). I prove this using the precise definition of limits of general functions and sequences. This theorem is also expected since a function at a particular value is equal to the limit as x approaches that particular value; thus we are just replacing the x term with the infinite sequence. 

#math #sequences #limits #education #calculus

Timestamps:

- Exercise 3:  Theorem 3: Sequence inside a function: 0:00
- Solution using the precise definition of limits: 1:27
- f is continuous at L so limit is same as plugging in function: 3:45
- Replace precise definition of a limit for a continuous function with the sequence: 5:54
- Precise definition of the limit is the same for our sequence: 7:29

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-2q1z3cwmj
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUgfK34Voi6BvrNqt4X4BV
- Notes: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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