LIMITS & CONTINUITYWhen learning calculus, a common first step is learning how to describe t
LIMITS & CONTINUITYWhen learning calculus, a common first step is learning how to describe the behavior of functions, for which limits & continuity form a particularly important foundation. The above are GIFs I’ve made in Mathematica related to these topics.I’ve been making these during breaks between writing sessions and they’ve been a lot of fun to create. If there are any other topics for which you’d like GIFs, let me know, and I might just do so. As always, I’m wishing you the best!GIF descriptions below the break.GIF 1: In this GIF, I’ve graphed a real function f(x) for x < c. From the plot’s left edge, a point travels along the function. As the point’s x-coordinate approaches c, its y-coordinate also approaches a value. This value is called “the Left-Hand (LH) Limit of f(x) as x approaches c." GIF 2: A limit such as the LH limit need not exist. For example, in this GIF, a function oscillates wildly as x approaches c from the left, such that f(x) never approaches any specific value. For this function, the LH limit doesn’t exist. GIF 3: A function’s behavior as x approaches c need not relate to the function’s value at x = c. f© may even be undefined. In this GIF, I return to our 1st GIF’s function & move f©’s value up-&-down while keeping the rest of f(x) in place. The LH Limit of f(x) at x = c is unchanged. GIF 4: This GIF sure is familiar: here I’ve graphed a real function f(x) for x > c. From the plot’s right edge, a point travels along the function. As the point’s x-coordinate approaches c, its y-coordinate this time approaches "the Right-Hand (RH) Limit of f(x) as x approaches c.” GIF 5: For many common functions, the LH & RH limits of a function at x = c are equal. In these cases, we call their shared value “the (Two-Sided) Limit of f(x) as x approaches c.” The GIF below combines the earlier LH & RH limit GIFs so that the limit of f(x) at x = c exists. GIF 6: The limit at x = c doesn’t exist if the LH and RH limits disagree or if either is nonexistent. In this GIF, the LH and RH limit GIFs have been shifted vertically apart before being combined. Thus, although the LH and RH limits at x = c both exist, the limit at x = c does not. GIF 7: Finally, as remarked earlier, the existence of a limit at x = c doesn’t mean the limit equals f© (if f© even exists at all). However, if the limit at x = c does happen to equal f©, we declare the function to be Continuous at x = c. This is a definition of continuity. -- source link
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