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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Welcome back ThegirlGodishelping🥰💙🫠 — missed you this week. Here are the solutions to the limit problems. 1. We need to evaluate the limit: _x 3 (x^3 - 27)/(x^2 - 9) Step 1: Substitute x=3 into the expression. (3^3 - 27)/(3^2 - 9) = (27 - 27)/(9 - 9) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Factor the numerator and the denominator. Use the difference of cubes formula a^3 - b^3 = (a-b)(a^2+ab+b^2) and the difference of squares formula a^2 - b^2 = (a-b)(a+b). x^3 - 27 = (x-3)(x^2 + 3x + 9) x^2 - 9 = (x-3)(x+3) Step 3: Rewrite the limit with the factored expressions and cancel out the common term (x-3). _x 3 ((x-3)(x^2 + 3x + 9))/((x-3)(x+3)) = _x 3 (x^2 + 3x + 9)/(x+3) Step 4: Substitute x=3 into the simplified expression. (3^2 + 3(3) + 9)/(3+3) = (9 + 9 + 9)/(6) = (27)/(6) Step 5: Simplify the fraction. (27)/(6) = (9)/(2) The limit is (9)/(2). 2. We need to evaluate the limit: _x 7 sqrt(x+2) - 3x-7 Step 1: Substitute x=7 into the expression. sqrt(7+2) - 37-7 = sqrt(9) - 30 = (3-3)/(0) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(x+2) + 3. _x 7 sqrt(x+2) - 3x-7 × sqrt(x+2) + 3sqrt(x+2) + 3 Step 3: Simplify the numerator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. (sqrt(x+2))^2 - 3^2 = (x+2) - 9 = x-7 Step 4: Rewrite the limit with the simplified numerator. _x 7 (x-7)/((x-7)(sqrt(x+2) + 3)) Step 5: Cancel out the common term (x-7). _x 7 (1)/(sqrt(x+2) + 3) Step 6: Substitute x=7 into the simplified expression. (1)/(sqrt(7+2) + 3) = (1)/(sqrt(9) + 3) = (1)/(3+3) = (1)/(6) The limit is (1)/(6). 3. We need to evaluate the limit: _x (4x^2 - 5x + 1)/(sqrt(9x^4 + 2x^2 - 6)) Step 1: Identify the highest power of x in the denominator. Inside the square root, the highest power is x^4, so sqrt(x^4) = x^2. Step 2: Divide every term in the numerator and the denominator by x^2. For terms inside the square root, divide by x^4. _x (4x^2)/(x^2) - (5x)/(x^2) + (1)/(x^2)sqrt((9x^4)/(x^4) + (2x^2)/(x^4) - (6)/(x^4)) Step 3: Simplify the terms. _x (4 - 5)/(x) + (1)/(x^2)sqrt(9 + (2)/(x^2) - (6)/(x^4)) Step 4: Apply the limit. As x , terms of the form (c)/(x^n) approach 0. (4 - 0 + 0)/(sqrt(9 + 0 - 0)) = (4)/(sqrt(9)) Step 5: Simplify the expression. (4)/(3) The limit is (4)/(3). 4. We need to evaluate the limit: _x 0 ((5x))/((3x)) Step 1: Rewrite (3x) in terms of (3x) and (3x). _x 0 ((5x))/((3x))(3x) = _x 0 ((5x) (3x))/((3x)) Step 2: Rearrange the terms to use the special limit _u 0 ( u)/(u) = 1. _x 0 ( ((5x))/(5x) · (3x)/((3x)) · (5x)/(3x) · (3x) ) Step 3: Simplify the expression and apply the limit properties. _x 0 ( ((5x))/(5x) ) · _x 0 ( (3x)/((3x)) ) · _x 0 ( (5)/(3) ) · _x 0 ( (3x) ) Step 4: Evaluate each limit. 1 · 1 · (5)/(3) · (0) = 1 · 1 · (5)/(3) · 1 = (5)/(3) The limit is (5)/(3). 5. There are two problems labeled '5'. I will solve both. First problem 5: _x 0 (1 - x)/(x^2) Step 1: This is a standard limit. We can use the identity 1 - x = 2 ^2((x)/(2)). _x 0 (2 ^2(x)/(2))x^2 Step 2: Rearrange the terms to use the special limit _u 0 ( u)/(u) = 1. _x 0 2 · ((x)/(2))x · ((x)/(2))x Step 3: Multiply and divide by (1)/(2) for each ((x)/(2))x term. _x 0 2 · ((x)/(2))(x)/(2) · 2 · ((x)/(2))(x)/(2) · 2 Step 4: Simplify and apply the limit. _x 0 2 · (1)/(2) · (1)/(2) · ( ((x)/(2))(x)/(2) ) · ( ((x)/(2))(x)/(2) ) = 2 · (1)/(4) · 1 · 1 = (1)/(2) Alternatively, using L'Hôpital's Rule (since substituting x=0 gives (0)/(0)): Step 1: Apply L'Hôpital's Rule. _x 0 (d)/(dx)(1 - x)(d)/(dx)(x^2) = _x 0 ( x)/(2x) Step 2: Rearrange the terms. _x 0 (1)/(2) · ( x)/(x) Step 3: Apply the special limit _x 0 ( x)/(x) = 1. (1)/(2) · 1 = (1)/(2) The limit is (1)/(2). Second problem 5: _x 0 (1 - x)/(x) Step 1: Substitute x=0 into the expression. (1 - 0)/(0) = (1 - 1)/(0) = (0)/(0) This is an indeterminate form, so we can use L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule. _x 0 (d)/(dx)(1 - x)(d)/(dx)(x) = _x 0 ( x)/(1) Step 3: Substitute x=0 into the simplified expression. (0) = 0 The limit is 0. Drop the next question.