This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the limits for each function: i) For f(x) = x^2: Step 1: Substitute f(x) and f(c) into the limit expression. _x c (x^2 - c^2)/(x-c) Step 2: Factor the numerator using the difference of squares formula, a^2 - b^2 = (a-b)(a+b). _x c ((x-c)(x+c))/(x-c) Step 3: Cancel out the (x-c) term, since x ≠ c as x c. _x c (x+c) Step 4: Substitute x=c into the simplified expression. c+c = 2c The limit is 2c. ii) For f(x) = (1)/(x): Step 1: Substitute f(x) and f(c) into the limit expression. _x c (1)/(x) - (1)/(c)x-c Step 2: Combine the fractions in the numerator. _x c (c-x)/(xc)x-c Step 3: Rewrite the expression by multiplying the numerator by the reciprocal of the denominator. _x c (c-x)/(xc(x-c)) Step 4: Factor out -1 from the numerator to match the term in the denominator. _x c (-(x-c))/(xc(x-c)) Step 5: Cancel out the (x-c) term. _x c (-1)/(xc) Step 6: Substitute x=c into the simplified expression. (-1)/(c · c) = (-1)/(c^2) The limit is -(1)/(c^2). iii) For f(x) = sqrt(x): Step 1: Substitute f(x) and f(c) into the limit expression. _x c sqrt(x) - sqrt(c)x-c Step 2: Multiply the numerator and denominator by the conjugate of the numerator, sqrt(x)+sqrt(c). _x c sqrt(x) - sqrt(c)x-c · sqrt(x)+sqrt(c)sqrt(x)+sqrt(c) Step 3: Apply the difference of squares formula (a-b)(a+b) = a^2 - b^2 to the numerator. _x c (sqrt(x))^2 - (sqrt(c))^2(x-c)(sqrt(x)+sqrt(c)) _x c (x-c)/((x-c)(sqrt(x)+c)) Step 4: Cancel out the (x-c) term. _x c (1)/(sqrt(x)+c) Step 5: Substitute x=c into the simplified expression. (1)/(sqrt(c)+c) = (1)/(2sqrt(c)) The limit is (1)/(2sqrt(c)). That's 2 down. 3 left today — send the next one.