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Exercise

$\lim_{x\to0^{-x}}\left(\sqrt{3x+\frac{9}{x^2}}\right)$

Step-by-step Solution

1

Evaluate the limit $\lim_{x\to{0^{-x}}}\left(\sqrt{3x+\frac{9}{x^2}}\right)$ by replacing all occurrences of $x$ by $0^{-x}$

$\sqrt{3\cdot 0^{-x}+\frac{9}{\left(0^{-x}\right)^2}}$
2

Simplify $\left(0^{-x}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $-x$ and $n$ equals $2$

$\sqrt{3\cdot 0^{-x}+\frac{9}{0^{- 2x}}}$
3

Multiply $-1$ times $2$

$\sqrt{3\cdot 0^{-x}+\frac{9}{0^{-2x}}}$

Final answer to the exercise

$\sqrt{3\cdot 0^{-x}+\frac{9}{0^{-2x}}}$

Try other ways to solve this exercise

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  • Solve using L'Hôpital's rule
  • Solve without using l'Hôpital
  • Solve using limit properties
  • Solve using direct substitution
  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
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Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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