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I find this sentiment somewhat surprising. Limits is incredibly easy to link to real physical things: movement, speed, acceleration. If speed is distance over

by hasenj 9y ago

I find this sentiment somewhat surprising. Limits is incredibly easy to link to real physical things: movement, speed, acceleration. If speed is distance over time, what's the speed of a moving object at a very specific instant?

Koshkin 9y ago
"Ma'am, you were driving at ninety miles an hour." "But, sir, I have only traveled twenty miles!"
jdmichal 9y ago
Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve. Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.
- hasenj 9y ago
  Will, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.
  - jdmichal 9y ago
    Ah, good point. I forgot about that problem with instantaneous speed.
  - hasenj 9y ago
    lol sorry for the typos! Wrote that from my mobile phone at night.
- Koshkin 9y ago
  I think these things have already been discussed to death some time around the year 1700, when they were called something like "mathematical analysis of infinitely small magnitudes" or some such; the difference between an "infinitely small magnitude" and zero was already clearly understood. The notion of a "limit" came only later, in an attempt to put this stuff on a firm logical foundation and was (and still is) seen more of a formality than something aiding our intuition in this area.

by hasenj 9y ago

Koshkin 9y ago

"Ma'am, you were driving at ninety miles an hour." "But, sir, I have only traveled twenty miles!"

jdmichal 9y ago

Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve. Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.

hasenj 9y ago
Will, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.
- jdmichal 9y ago
  Ah, good point. I forgot about that problem with instantaneous speed.
- hasenj 9y ago
  lol sorry for the typos! Wrote that from my mobile phone at night.
Koshkin 9y ago
I think these things have already been discussed to death some time around the year 1700, when they were called something like "mathematical analysis of infinitely small magnitudes" or some such; the difference between an "infinitely small magnitude" and zero was already clearly understood. The notion of a "limit" came only later, in an attempt to put this stuff on a firm logical foundation and was (and still is) seen more of a formality than something aiding our intuition in this area.