lim _(substack(x arrow 0 0)) (5^x-3^x-2^x+1)/(x^2)

To find the limit of the given expression as approaches 0, we can use the Taylor series expansion for each term in the numerator and the denominator.The Taylor series expansion for around is: The Taylor series expansion for around is: The Taylor series expansion for around is: The Taylor series expansion for around is: Now, let's substitute these expansions into the given expression: Simplifying the numerator, we get: Now, we can cancel out the common factor of in the numerator and denominator: Finally, substituting into the expression, we get: Since the denominator is zero, the limit is undefined. Therefore, the correct answer is that the limit does not exist.