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do you mean $\displaystyle \frac {\sqrt{1 + u} - \sqrt{1 + u^2}}{u - 1}$ ?
if so, multiply by $\displaystyle \frac {\sqrt{1 + u} + \sqrt{1 + u^2}}{\sqrt{1 + u} + \sqrt{1 + u^2}}$ (the conjugate of the numerator over itself) and simplify as far as possible before taking the limit. you should be able to cancel out the denominator and hence remove the problem
Rationalize the numerator:
$\displaystyle \lim_{u \to 1} \left( \frac{\sqrt{1+u} - \sqrt{1+u^2}}{u-1} \cdot \frac{{\color{red}\sqrt{1+u} + \sqrt{1+u^2}}}{{\color{red}\sqrt{1+u} + \sqrt{1+u^2}}}\right)$
Simplify the numerator (notice it is a difference of squares and don't worry about the denominator .. you'll see why)