. lim _(xarrow 0)(sqrt [4](x^4+1)-sqrt (x^2+1))/(x^2) b) r->0

To find the limit of the given expression as x approaches 0, we can use L'Hopital's rule.L'Hopital's rule states that if the limit of a function as x approaches a certain value is in the form of 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and denominator separately and then taking the limit of the ratio of the derivatives.In this case, the given expression is in the form of 0/0 as x approaches 0. Therefore, we can apply L'Hopital's rule.Taking the derivative of the numerator, we get: Simplifying, we get: Taking the derivative of the denominator, we get: Now, we can apply L'Hopital's rule to find the limit: Simplifying, we get: As x approaches 0, the numerator approaches 0, and the denominator approaches 2. Therefore, the limit is 0.So, the answer is 0.