Not to be pedantic, but the top equation is wrong. If x is a tiny bit above 8, then the fraction is positive, but if x is a tiny bit below 8, then the fraction is negative. While the absolute value approaches infinity, the actual function is near both positive and negative infinity. So the limit doesn't exist.
If the bottom is changed to (x-8)^2 then it's correct. Or you can take a one-sided limit by putting a tiny + or - to the right of the 8. :)
Funny, I was just thinking the same thing - as x approaches 8, from where is it coming?
Although I thought it was still pretty hilaaarious. How about this one?
A bunch of functions were hanging out at the q-bar (haha, yeah, but we're not at the punchline yet). Then all of a sudden, a function cries out, "Oh sh*t! Look out, Differentiation is coming!" And pssw! - everyone scrams, and when the dust finally lifts, all the functions have vanished. Except e^x, who hasn't moved.
Differentiation enters the bar. He looks around, and sees a lone e^x chilling on a bar stool sipping his beer. Differentiation goes over and says, "Do you not know who I am? I'm Differentiation!" e^x looks up and says, "I know," and turns back to his drink. Differentiation says, "Why, you sure got guts for a function. Who on R are you?" e^x says, "I'm e^x. So see, there's nothing you can do to me." He thumbs his nose at him, and continues drinking. "Weeell," Differentiation says with a smile, "There is one thing you're forgetting. Who said I differentiate along x?"
5 comments:
Not to be pedantic, but the top equation is wrong. If x is a tiny bit above 8, then the fraction is positive, but if x is a tiny bit below 8, then the fraction is negative. While the absolute value approaches infinity, the actual function is near both positive and negative infinity. So the limit doesn't exist.
If the bottom is changed to (x-8)^2 then it's correct. Or you can take a one-sided limit by putting a tiny + or - to the right of the 8. :)
No! No! Be pedantic! (But I am not convinced that such an artifact could not surface in a junk drawer.)
I debated whether pedantry was acceptable on this blog, but then I rembered your post on greek typesetting in Mr Eliot's Sunday Morning Service :)
Funny, I was just thinking the same thing - as x approaches 8, from where is it coming?
Although I thought it was still pretty hilaaarious. How about this one?
A bunch of functions were hanging out at the q-bar (haha, yeah, but we're not at the punchline yet). Then all of a sudden, a function cries out, "Oh sh*t! Look out, Differentiation is coming!" And pssw! - everyone scrams, and when the dust finally lifts, all the functions have vanished. Except e^x, who hasn't moved.
Differentiation enters the bar. He looks around, and sees a lone e^x chilling on a bar stool sipping his beer. Differentiation goes over and says, "Do you not know who I am? I'm Differentiation!" e^x looks up and says, "I know," and turns back to his drink. Differentiation says, "Why, you sure got guts for a function. Who on R are you?" e^x says, "I'm e^x. So see, there's nothing you can do to me." He thumbs his nose at him, and continues drinking.
"Weeell," Differentiation says with a smile, "There is one thing you're forgetting. Who said I differentiate along x?"
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