Evaluate the limit: lim (t->0) x / (3 - sqrt(x+9))

The problem asks to evaluate the limit: _t 0 (x)/(3 - sqrt(x+9)) There appears to be a variable mismatch, as the limit is for t 0 but the expression contains x. Assuming the limit should be with respect to x, we will evaluate: _x 0 (x)/(3 - sqrt(x+9)) Step 1: Attempt direct substitution. Substitute x=0 into the expression: (0)/(3 - sqrt(0+9)) = (0)/(3 - sqrt(9)) = (0)/(3 - 3) = (0)/(0) This is an indeterminate form, so we need to manipulate the expression. Step 2: Rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is 3 + sqrt(x+9). _x 0 (x)/(3 - sqrt(x+9)) × 3 + sqrt(x+9)3 + sqrt(x+9) Step 3: Simplify the expression. Using the difference of squares formula (a-b)(a+b) = a^2 - b^2 for the denominator: (3 - sqrt(x+9))(3 + sqrt(x+9)) = 3^2 - (sqrt(x+9))^2 = 9 - (x+9) = 9 - x - 9 = -x The expression becomes: _x 0 x(3 + sqrt(x+9))-x Step 4: Cancel common factors. Since x 0, x is not exactly zero, so we can cancel x from the numerator and denominator: _x 0 -(3 + sqrt(x+9)) Step 5: Evaluate the limit by direct substitution. Now substitute x=0 into the simplified expression: -(3 + sqrt(0+9)) -(3 + sqrt(9)) -(3 + 3) -(6) -6 The final answer is -6. Send me the next one 📸