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Find a limit

goody

New member
Mar 31, 2020
13
Good evening! Could anybody help me with this limit?

limit.png
I have a problem when there are two variables. The only thing I did was that:

limit2.png,
but I don't know if it was helpful. Thank you!
 

Cbarker1

Active member
Jan 8, 2013
227
MN
What is the variable in the limit symbol?

Cbarker1
 

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
613
North Texas
Good evening! Could anybody help me with this limit?

View attachment 10126
I have a problem when there are two variables. The only thing I did was that:

View attachment 10129,
but I don't know if it was helpful. Thank you!
actually, factoring out $(x-a)$ in the numerator yields ...

$\dfrac{(x-a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + ... + xa^{n-2} + a^{n-1} - na^{n-1})}{(x-a)^2}$

I played with the algebra to see if I could get the other $(x-a)$ factor in the denominator to divide out ... no joy, yet.

So, starting again ...

$\displaystyle \lim_{x \to a} \dfrac{(x^n-a^n) - na^{n-1}(x-a)}{(x-a)^2}$

using L'Hopital ...

$\displaystyle \lim_{x \to a} \dfrac{nx^{n-1} - na^{n-1}}{2(x-a)}$

$\displaystyle \lim_{x \to a} \dfrac{n(x^{n-1} - a^{n-1})}{2(x-a)}$

L'Hopital again ...

$\displaystyle \lim_{x \to a} \dfrac{n(n-1)(x^{n-2})}{2} = \dfrac{n(n-1)(a^{n-2})}{2}$