Square root limit

[johnjones]

lim t -> 0 of this function:

[sqroot(t^2 + 1) - 1] / t^2

I put t for zero into the equation, but I can't have 0 in the denominator. What next? I can't divide by zero. :?:

 

[wjm11]

lim t -> 0 of this function:

[sqroot(t^2 + 1) - 1] / t^2

Try multiplying by

[sqroot(t^2 + 1) + 1]/ [sqroot(t^2 + 1) + 1]


Is that enough to get you started?

 

[pka]

One other point. In doing limits, we never 'put' the limiting value into the function. The point is: what happens near the value? Not at the value!

 

[johnjones]

lim t -> 0 of this function:

[sqroot(t^2 + 1) - 1] / t^2

Try multiplying by

[sqroot(t^2 + 1) + 1]/ [sqroot(t^2 + 1) + 1]


Is that enough to get you started?

I got t^2/t^2(sroot[t^2 + 1]+1)

Now when I try to put zero, I'm still dividing by zero.

 

[pka]

Can someone please tell me where the idea of "putting in the limiting value" come from. Has our calculus education come to such a low point? Do calculus teachers actually teach this?

 

[wjm11]

I got t^2/t^2(sroot[t^2 + 1]+1)

Good! Now you can cancel the t^2 terms, and you’re left with

1/(sroot[t^2 + 1]+1)

Now what happens when you do the substitution?

BTW, what pka is correctly pointing out is what a “limit” really is all about. There are only certain circumstances that allow simple substitution to arrive at the answer. What a limit is telling you is NOT the value of the function at a point, but rather the behavior of the function in a neighborhood (close proximity) of the point.

 

[johnjones]

Can someone please tell me where the idea of "putting in the limiting value" come from. Has our calculus education come to such a low point? Do calculus teachers actually teach this?

Seems like everyone does it these days, even textbooks. Plug the limit in, then stop writing lim x-> 0 b/c we are subbing into the equation. How would you solve it w/o plugging it in then, pka? I mean, any calculus question in general. Like say you don't have a calculator and you have to solve a complicated limit problem?

 

[tkhunny]

Seems like everyone does it these days, even textbooks. Plug the limit in, then stop writing lim x-> 0 b/c we are subbing into the equation. How would you solve it w/o plugging it in then, pka? I mean, any calculus question in general. Like say you don't have a calculator and you have to solve a complicated limit problem?

Ummm...When were calculators invented? Complicated limit problems existed (and were solved) long before that time.

My college days were difficult on this point. My fellow students were limit pluggers - to an individual. I refused to follow this weak convention that entirely short-circuited the limit concept. The error in thought has, indeed, been around for a few years.

 

[pka]

johnjones,
You asked me “How would you solve it w/o plugging it in then, pka?”
Here is my reply. You can use it as an example.
http://www.freemathhelp.com/forum/viewtopic.php?p=29132#29132