Evaluate the limit: lim x->0 (sqrt(x2+9)-3)/x2 and lim x->1 (e^(x-1)-1)/(x-1)2

Step 1: Evaluate the limit by direct substitution. Substituting x=0 into the expression gives: sqrt(0^2+9)-30^2 = sqrt(9)-30 = (3-3)/(0) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Multiply by the conjugate. 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: Simplify the expression. Cancel out the x^2 term from the numerator and denominator (since x 0, x ≠ 0): = _x 0 (1)/(sqrt(x^2+9)+3) Step 4: Evaluate the limit by direct substitution. 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). Step 1: Evaluate the limit by direct substitution. Substituting x=1 into the expression gives: e^1-1-1(1-1)^2 = (e^0-1)/(0^2) = (1-1)/(0) = (0)/(0) This is an indeterminate form, so we can apply L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule. Take the derivative of the numerator and the denominator: Numerator: (d)/(dx)(e^x-1-x) = e^x-1 · 1 - 1 = e^x-1-1 Denominator: (d)/(dx)((x-1)^2) = 2(x-1) · 1 = 2(x-1) The limit becomes: _x 1 e^x-1-12(x-1) Step 3: Evaluate the limit again by direct substitution. Substituting x=1 into the new expression gives: 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. Take the derivative of the new numerator and denominator: Numerator: (d)/(dx)(e^x-1-1) = e^x-1 · 1 - 0 = e^x-1 Denominator: (d)/(dx)(2(x-1)) = 2 · 1 = 2 The limit becomes: _x 1 e^x-12 Step 5: Evaluate the limit by direct substitution. Substitute x=1 into the expression: = e^1-12 = (e^0)/(2) = (1)/(2) The limit is (1)/(2). Step 1: Evaluate the limit by direct substitution. Substituting x=4 into the expression gives: (4-4)/(sqrt(4)-2) = (0)/(2-2) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Factor the numerator. Recognize that the numerator x-4 can be written as a difference of squares, (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: Simplify the expression. Cancel out the common term (sqrt(x)-2) from the numerator and denominator (since x 4, sqrt(x) ≠ 2): = _x 4 (sqrt(x)+2) Step 4: Evaluate the limit by direct substitution. Substitute x=4 into the simplified expression: = sqrt(4)+2 = 2+2 = 4 The limit is 4.