Limits by conjugates
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When we want to evaluate a limit of a function, it is sometimes useful to know the rationalizing the numerator/denominator of the fraction in the function itself.
Prove \( \displaystyle \lim_{x\to\infty} \left( \sqrt{x^2+2x} - x \right) = 1 \).
The conjugate of \( \sqrt{x^2+2x} - x \) is \( \sqrt{x^2+2x} + x \), so we multiply the limit by the function \( \dfrac{\sqrt{x^2+2x} + x}{\sqrt{x^2+2x} + x} \) to get
\[ \begin{eqnarray} \lim_{x\to\infty}\left( \sqrt{x^2+2x} - x \right) & = &\lim_{x\to\infty} \left( \sqrt{x^2+2x} - x \right) \cdot\dfrac{\sqrt{x^2+2x} + x}{\sqrt{x^2+2x} + x} \\ &=& \lim_{x\to\infty}\dfrac{ \left( \sqrt{x^2+2x} - x \right) \left( \sqrt{x^2+2x} + x \right)} {\sqrt{x^2+2x} + x} \\ &=& \lim_{x\to\infty}\dfrac{ (x^2+2x) - x^2} {\sqrt{x^2+2x} + x} \\ &=& \lim_{x\to\infty}\dfrac{ 2x} {\sqrt{x^2+2x} + x} \\ &=& \lim_{x\to\infty}\dfrac{ 2} {\sqrt{1+2/x} + 1} \\ &=& \dfrac2{\sqrt{1+0} + 1} = 1 \end{eqnarray}\]
\[ \displaystyle \lim_{x \to \infty} \left ( \sqrt{x^4 + ax^3 + 3x^2 + bx + 2} - \sqrt{x^4 + 2x^3 - cx^2 + 3x - d} \ \right ) \]
The limit above equals to 4 for constants \(a,b,c,d\).
What is the value of the expression below?
\[ a + c + \left ( b - \frac {4d}{5} \right ) \left ( a - \frac {10} c \right ) \]