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.
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Here are the solutions to the limit problems. Problem 1: _x 0 sqrt(x^2+9)-3x^2 Step 1: Substitute x=0 into the expression. sqrt(0^2+9)-30^2 = sqrt(9)-30 = (3-3)/(0) = (0)/(0) This is an indeterminate form. Step 2: Multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(x^2+9)+3. _x 0 sqrt(x^2+9)-3x^2 × sqrt(x^2+9)+3sqrt(x^2+9)+3 = _x 0 (sqrt(x^2+9))^2 - 3^2x^2(sqrt(x^2+9)+3) = _x 0 (x^2+9-9)/(x^2(sqrt(x^2+9)+3)) = _x 0 (x^2)/(x^2(sqrt(x^2+9)+3)) Step 3: Cancel out the x^2 term from the numerator and denominator. = _x 0 (1)/(sqrt(x^2+9)+3) Step 4: Substitute x=0 into the simplified expression. = (1)/(sqrt(0^2+9)+3) = (1)/(sqrt(9)+3) = (1)/(3+3) = (1)/(6) The limit is (1)/(6). Problem 2: _x 1 e^x-1-x(x-1)^2 Step 1: Substitute x=1 into the expression. e^1-1-1(1-1)^2 = (e^0-1)/(0^2) = (1-1)/(0) = (0)/(0) This is an indeterminate form, so we apply L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule by taking the derivative of the numerator and the denominator. Let f(x) = e^x-1-x and g(x) = (x-1)^2. f'(x) = (d)/(dx)(e^x-1-x) = e^x-1-1 g'(x) = (d)/(dx)((x-1)^2) = 2(x-1) _x 1 e^x-1-12(x-1) Step 3: Substitute x=1 again. e^1-1-12(1-1) = (e^0-1)/(0) = (1-1)/(0) = (0)/(0) This is still an indeterminate form, so we apply L'Hôpital's Rule again. Step 4: Apply L'Hôpital's Rule a second time. Let f_2(x) = e^x-1-1 and g_2(x) = 2(x-1). f_2'(x) = (d)/(dx)(e^x-1-1) = e^x-1 g_2'(x) = (d)/(dx)(2(x-1)) = 2 _x 1 e^x-12 Step 5: Substitute x=1 into the expression. = e^1-12 = (e^0)/(2) = (1)/(2) The limit is (1)/(2). Problem 3: _x 4 (x-4)/(sqrt(x)-2) Step 1: Substitute x=4 into the expression. (4-4)/(sqrt(4)-2) = (0)/(2-2) = (0)/(0) This is an indeterminate form. Step 2: Factor the numerator using the difference of squares formula, a^2-b^2 = (a-b)(a+b). We can write x-4 as (sqrt(x))^2 - 2^2. x-4 = (sqrt(x)-2)(sqrt(x)+2) Substitute this back into the limit expression: _x 4 (sqrt(x)-2)(sqrt(x)+2)sqrt(x)-2 Step 3: Cancel out the common term (sqrt(x)-2) from the numerator and denominator. = _x 4 (sqrt(x)+2) Step 4: Substitute x=4 into the simplified expression. = sqrt(4)+2 = 2+2 = 4 The limit is 4.