Limits with roots and parameters

Task number: 2979

Depending on the parameters \(m,n\in \mathbb N\) a \(a,b\in\mathbb R\) determine the limits:

  • Variant 1

    \(\displaystyle \lim_{x\to 0} \frac{(1+mx)^n-(1+nx)^m}{x^2} \)

  • Variant 2

    \(\displaystyle \lim_{x\to \infty} \sqrt{(x+a)(x+b)}-x \)

  • Variant 3

    \(\displaystyle \lim_{x\to 0} \frac{\sqrt[n]{1 + x}-1}x \)

  • Variant 4

    \(\displaystyle \lim_{x\to 1} \frac{\sqrt[m]{x}-1}{\sqrt[n]{x}-1} \)

  • Variant 5

    \(\displaystyle \lim_{x\to 0} \frac{\sqrt[m]{1 + ax}-\sqrt[n]{1 + bx}}x \)

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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