How to find the limit of the function mentioned below?

OmarMohamedKhallaf

New member
Joined
Nov 22, 2019
Messages
22
\(\displaystyle \lim\limits_{x \to 1}\frac{x^{\frac{13}{2}}-x^{\frac{1}{2}}}{x^{\frac{7}{2}}-x^{\frac{1}{2}}}\)
What I did:
\(\displaystyle \lim\limits_{x \to 1}\frac{x^{\frac{13}{2}}-x^{\frac{1}{2}}}{x^{\frac{7}{2}}-x^{\frac{1}{2}}}=\lim\limits_{x \to 1}\frac{x^{\frac{1}{2}}(x^{13}-1)}{x^{\frac{1}{2}}(x^{7}-1)}=\lim\limits_{x \to 1}\frac{x^{13}-1^{13}}{x^{7}-1^{7}}=\frac{13}{7}\times 1^{13-7}=\frac{13}{7}\)
But the answer seems to be 2. Where did I go wrong?
 

topsquark

Full Member
Joined
Aug 27, 2012
Messages
641
\(\displaystyle \lim\limits_{x \to 1}\frac{x^{\frac{13}{2}}-x^{\frac{1}{2}}}{x^{\frac{7}{2}}-x^{\frac{1}{2}}}\)
Take a closer look at when you factored \(\displaystyle x^{13/2}\)
\(\displaystyle x^{13/2} = x^{1/2} \cdot x^{13/2 - 1/2} = x^{1/2} \cdot x^{12/2} = x^{1/2} \cdot x^6\)

-Dan
 
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