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b) lim_(x rarr1)(sqrt(3x^(2)+sqrtx)-sqrt(5x^(2)-sqrtx))/(sqrt7x^(2)+2sqrtx-sqrt(11x^(2)-2sqrtx))

b) lim⁡x→13x2+x−5x2−x7x2+2x−11x2−2x \lim _{x \rightarrow 1} \frac{\sqrt{3 x^{2}+\sqrt{x}}-\sqrt{5 x^{2}-\sqrt{x}}}{\sqrt{7} x^{2}+2 \sqrt{x}-\sqrt{11 x^{2}-2 \sqrt{x}}}

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Full solution

Q. b) lim⁡x→13x2+x−5x2−x7x2+2x−11x2−2x \lim _{x \rightarrow 1} \frac{\sqrt{3 x^{2}+\sqrt{x}}-\sqrt{5 x^{2}-\sqrt{x}}}{\sqrt{7} x^{2}+2 \sqrt{x}-\sqrt{11 x^{2}-2 \sqrt{x}}}
  1. Simplify Inside the Limit: First, let's simplify the expression inside the limit. We can use the conjugate to rationalize the numerator. The conjugate of the numerator is 3x2+x+5x2−x\sqrt{3x^2 + \sqrt{x}} + \sqrt{5x^2 - \sqrt{x}}.
  2. Multiply by Conjugate: Multiply the numerator and denominator by the conjugate of the numerator: (3x2+x−5x2−x)∗(3x2+x+5x2−x)(7x2+2x−11x2−2x)∗(3x2+x+5x2−x)\frac{(\sqrt{3x^2 + \sqrt{x}} - \sqrt{5x^2 - \sqrt{x}}) * (\sqrt{3x^2 + \sqrt{x}} + \sqrt{5x^2 - \sqrt{x}})}{(\sqrt{7x^2 + 2\sqrt{x}} - \sqrt{11x^2 - 2\sqrt{x}}) * (\sqrt{3x^2 + \sqrt{x}} + \sqrt{5x^2 - \sqrt{x}})}
  3. Simplify Numerator: Simplify the numerator using the difference of squares formula: \newline(3x2+x)−(5x2−x)(3x^2 + \sqrt{x}) - (5x^2 - \sqrt{x})
  4. Combine Like Terms: Combine like terms in the numerator: −2x2+2x-2x^2 + 2\sqrt{x}
  5. Simplify Denominator: Now, let's simplify the denominator. We can't directly apply the difference of squares because of the middle term 2x2\sqrt{x}. So, we'll just multiply the conjugate by the denominator and see what we get: (7x2+2x−11x2−2x)×(3x2+x+5x2−x)(\sqrt{7x^2 + 2\sqrt{x}} - \sqrt{11x^2 - 2\sqrt{x}}) \times (\sqrt{3x^2 + \sqrt{x}} + \sqrt{5x^2 - \sqrt{x}})
  6. Evaluate Limit: Oops, I made a mistake. We don't need to multiply the denominator by the conjugate because we're looking for the limit as xx approaches 11. Let's just plug in x=1x = 1 and see if we get an indeterminate form.

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