Evaluate the validity of applying L’Hospital’s Rule to

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

\displaystyle \lim_{x\to0}\dfrac{\sqrt{x^{2}+1}-1}{|x|}

. Which statement is most accurate? (a) L’Hospital’s Rule cannot be applied because the limit is not of the form

0/0

\infty/\infty

. (b) L’Hospital’s Rule can be applied because both functions are differentiable on an open interval containing

0

except possibly at

0

, and

\dfrac{d}{dx}|x|\neq0

for

x\neq0

in that interval. (c) L’Hospital’s Rule can be applied solely because both numerator and denominator approach

0

. (d) L’Hospital’s Rule cannot be applied because

|x|

is not differentiable at

x=0