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Evaluate the limit: lim (x->0) [sqrt(2x+3) - sqrt(3)] / x

Mathematics

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Answer

33\frac{\sqrt{3}}{3}

Step-by-step solution

7 steps

  1. Substitute x=0x=0 into the expression to check for an indeterminate form.

    2(0)+3−30=3−30=00\frac{\sqrt{2(0)+3} - \sqrt{3}}{0} = \frac{\sqrt{3} - \sqrt{3}}{0} = \frac{0}{0}

    This is an indeterminate form, so we need to simplify the expression.

  2. Multiply the numerator and the denominator by the conjugate of the numerator, which is 2x+3+3\sqrt{2x+3} + \sqrt{3}.

    lim⁡x→02x+3−3x×2x+3+32x+3+3\lim_{x \to 0} \frac{\sqrt{2x+3} - \sqrt{3}}{x} \times \frac{\sqrt{2x+3} + \sqrt{3}}{\sqrt{2x+3} + \sqrt{3}}

  3. Apply the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2 to the numerator.

    lim⁡x→0(2x+3)2−(3)2x(2x+3+3)\lim_{x \to 0} \frac{(\sqrt{2x+3})^2 - (\sqrt{3})^2}{x(\sqrt{2x+3} + \sqrt{3})} lim⁡x→0(2x+3)−3x(2x+3+3)\lim_{x \to 0} \frac{(2x+3) - 3}{x(\sqrt{2x+3} + \sqrt{3})}

  4. Simplify the numerator.

    lim⁡x→02xx(2x+3+3)\lim_{x \to 0} \frac{2x}{x(\sqrt{2x+3} + \sqrt{3})}

  5. Cancel out the common factor xx from the numerator and denominator.

    lim⁡x→022x+3+3\lim_{x \to 0} \frac{2}{\sqrt{2x+3} + \sqrt{3}}

  6. Substitute x=0x=0 into the simplified expression.

    22(0)+3+3=23+3\frac{2}{\sqrt{2(0)+3} + \sqrt{3}} = \frac{2}{\sqrt{3} + \sqrt{3}} 223\frac{2}{2\sqrt{3}}

  7. Simplify the result.

    13\frac{1}{\sqrt{3}}

    To rationalize the denominator, multiply the numerator and denominator by 3\sqrt{3}.

    13×33=33\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}

    The final answer is 33\frac{\sqrt{3}}{3}.

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