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Exercise

$\lim_{x\to1\:y\:\to\:0}\left(\sqrt{\frac{x+y-1}{x^2+2y+4}}\right)$

Step-by-step Solution

1

Evaluate the limit $\lim_{y\to0}\left(\sqrt{\frac{x+y-1}{x^2+2y+4}}\right)$ by replacing all occurrences of $y$ by $0$

$x\sqrt{\frac{x+0-1}{x^2+2\cdot 0+4}}$
2

Subtract the values $0$ and $-1$

$x\sqrt{\frac{x-1}{x^2+2\cdot 0+4}}$
3

Multiply $2$ times $0$

$x\sqrt{\frac{x-1}{x^2+0+4}}$
4

Add the values $0$ and $4$

$x\sqrt{\frac{x-1}{x^2+4}}$
5

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$x\frac{\sqrt{x-1}}{\sqrt{x^2+4}}$

Final answer to the exercise

$x\frac{\sqrt{x-1}}{\sqrt{x^2+4}}$

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
  • FOIL Method
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Main Topic: Limits

In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value.

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