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Find lim_(h rarr0)(6sqrt(9+h)-6sqrt9)/(h). Choose 1 answer: (A) 1 (B) 6 (C) 18 (D) The limit doesn't exist

Find lim⁡h→069+h−69h \lim _{h \rightarrow 0} \frac{6 \sqrt{9+h}-6 \sqrt{9}}{h} .\newlineChoose 11 answer:\newline(A) 11\newline(B) 66\newline(C) 1818\newline(D) The limit doesn't exist

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Full solution

Q. Find lim⁡h→069+h−69h \lim _{h \rightarrow 0} \frac{6 \sqrt{9+h}-6 \sqrt{9}}{h} .\newlineChoose 11 answer:\newline(A) 11\newline(B) 66\newline(C) 1818\newline(D) The limit doesn't exist
  1. Identify Problem: Identify the limit problem and apply the conjugate to simplify the expression.\newlineMultiply the numerator and denominator by the conjugate of the numerator: 69+h+69.6\sqrt{9+h}+6\sqrt{9}.
  2. Apply Conjugate: Perform the multiplication in the numerator: (69+h−69)(69+h+69)(6\sqrt{9+h}-6\sqrt{9})(6\sqrt{9+h}+6\sqrt{9}).
  3. Multiply Numerator and Denominator: Use the difference of squares formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). The numerator becomes: 36(9+h)−36×936(9+h) - 36\times 9.
  4. Use Difference of Squares: Simplify the numerator: 36×9+36h−36×936\times 9 + 36h - 36\times 9.
  5. Simplify Numerator: Cancel out 36×936\times 9 terms in the numerator to get 36h36h.
  6. Cancel Terms: The expression simplifies to 36h/h36h/h.
  7. Final Simplification: Cancel out the hh terms to get 3636.

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