Park Schwartz
04/01/2023 · Elementary School
\( \lim _{x \rightarrow \infty } (\frac{x(\sqrt {x+1}-\sqrt {x}}{\sqrt {2x}-\sqrt {x}}) \)
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Step-by-step Solution
Calculate or simplify the expression \( \lim_{x\rightarrow \infty} (\frac{x(\sqrt{x+1}-\sqrt{x}}{\sqrt{2x}-\sqrt{x}})) \).
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{x\rightarrow +\infty}\left(\frac{x\left(\sqrt{x+1}-\sqrt{x}\right)}{\sqrt{2x}-\sqrt{x}}\right)\)
- step1: Multiply the terms:
\(\lim _{x\rightarrow +\infty}\left(\frac{\left(\sqrt{x+1}-\sqrt{x}\right)x}{\sqrt{2x}-\sqrt{x}}\right)\)
- step2: Rewrite the expression:
\(\lim _{x\rightarrow +\infty}\left(\frac{\left(\sqrt{1+\frac{1}{x}}-1\right)x^{\frac{3}{2}}}{\left(\sqrt{2}-1\right)x^{\frac{1}{2}}}\right)\)
- step3: Simplify:
\(\lim _{x\rightarrow +\infty}\left(\frac{\left(\sqrt{1+\frac{1}{x}}-1\right)x}{\sqrt{2}-1}\right)\)
- step4: Rewrite the expression:
\(\frac{\lim _{x\rightarrow +\infty}\left(\left(\sqrt{1+\frac{1}{x}}-1\right)x\right)}{\lim _{x\rightarrow +\infty}\left(\sqrt{2}-1\right)}\)
- step5: Calculate:
\(\frac{\frac{1}{2}}{\lim _{x\rightarrow +\infty}\left(\sqrt{2}-1\right)}\)
- step6: Calculate:
\(\frac{\frac{1}{2}}{\sqrt{2}-1}\)
- step7: Multiply by the reciprocal:
\(\frac{1}{2}\times \frac{1}{\sqrt{2}-1}\)
- step8: Multiply the fractions:
\(\frac{1}{2\left(\sqrt{2}-1\right)}\)
- step9: Multiply:
\(\frac{1}{2\sqrt{2}-2}\)
- step10: Multiply by the Conjugate:
\(\frac{2\sqrt{2}+2}{\left(2\sqrt{2}-2\right)\left(2\sqrt{2}+2\right)}\)
- step11: Multiply the numbers:
\(\frac{2\sqrt{2}+2}{4}\)
- step12: Factor the expression:
\(\frac{\left(\sqrt{2}+1\right)\times 2}{2\times 2}\)
- step13: Reduce the fraction:
\(\frac{\sqrt{2}+1}{2}\)
La expresión dada se simplifica a \( \frac{\sqrt{2}+1}{2} \) cuando se toma el límite como \( x \) se acerca a infinito.
Quick Answer
El límite de la expresión es \( \frac{\sqrt{2}+1}{2} \).
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