Standard Limits

The following limits do need any further proof to be used. Make sure to say that they are standard limits in working out.

1. ${\lim }_{x\to \infty }\frac{1}{x^p}=0,(p>0)$

2. ${\lim }_{x\to \infty }{r}^x=0,(|r|<1)$

3. ${\lim }_{x\to \infty }{a}^{\frac{1}{x}}=1,(a>0)$

4. ${\lim }_{x\to \infty }{x}^{\frac{1}{x}}=1$

5. ${\lim }_{x\to \infty }\frac{a^x}{x!}=0,(a\in \mathbb{R})$

6. ${\lim }_{x\to \infty }\frac{\log (x)}{x^p},(p>0)$

7. ${\lim }_{x\to \infty }(1+\frac{a}{n}{)}^n=e^a,(a\in \mathbb{R})$

8. ${\lim }_{x\to \infty }\frac{n^p}{a^n}=0,(p\in \mathbb{R},a>1)$