Definitions [2]
If\[\operatorname*{lim}_{x\to a}\frac{f(x)}{g(x)}\] is of the indeterminate form \[\frac{0}{0}\], then factorise f(x) and g(x) and cancel the common factors to evaluate the limit.
If we get \[\frac{0}{0}\] form and the numerator or denominator or both have a radical sign, then rationalise and substitute the limit.
Formulae [1]
1. \[\lim_{x\to0}\left(\frac{e^{x}-1}{x}\right)=\log e=1\]
2. \[\lim_{x\to0}\left(\frac{a^{x}-1}{x}\right)=\log a(a>0,a\neq1)\]
3. \[\lim_{x\to0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}\]
4. \[\lim_{x\to0}\left(\frac{\log\left(1+x\right)}{x}\right)=1\]
5. \[\lim_{x\to0}\left(\frac{e^{px}-1}{px}\right)=1,\] (p constant)
6. \[\lim_{x\to0}\left(\frac{a^{px}-1}{px}\right)=\log a,\] (p constant)
7. \[\lim_{x\to\infty}a^{x}= \begin{cases} 0 & , & \mathrm{if}-1<a<1 \\ 1 & , & \mathrm{if}a=1 \\ \infty & , & \mathrm{if}a>1 & \end{cases}\]
8. \[\lim_{x\to0}\frac{\log\left(1+\mathrm{k}x\right)}{x}=\mathrm{k},\mathrm{k}\in\mathrm{R}\]
Key Points
| No. | Rule | Limit Law |
|---|---|---|
| i | Sum | \[\lim_{x\to a}\left(f+g\right)x=\lim_{x\to a}f\left(x\right)+\lim_{x\to a}g\left(x\right)\] |
| ii | Difference | \[\lim_{x\to a}\left(f-g\right)x=\lim_{x\to a}f\left(x\right)-\lim_{x\to a}g\left(x\right)\] |
| iii | Product |
\[\lim_{x\to a}\left[f(x)\cdot g(x)\right]=\lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x)\] |
| iv | Constant multiple | \[\lim_{x\to a}[c\cdot f(x)]=c\cdot\lim_{x\to a}f(x)\] |
| v | Quotient |
\[\lim_{x\to a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim_{x\to a}f\left(x\right)}{\lim_{x\to a}g\left(x\right)}\] where \[\lim_{x\to a}g\left(x\right)\neq0\] |
| vi | Function of function | \[\lim_{x\to a}\mathrm{f}\left[\mathrm{g}(x)\right]=\mathrm{f}\left[\lim_{x\to a}\mathrm{g}\left(x\right)\right]=\mathrm{f}(\mathrm{m})\] |
| vii | Sum with constant | (\lim [f(x)+k] = \lim f(x) + k = l + k) |
| viii | Logarithmic | \[\lim_{x\to a}\log\left[\mathrm{f}(x)\right]=\log\left[\lim_{x\to a}\mathrm{f}(x)\right]=\log l\] |
| ix | Power | \[\lim_{x\to a}[\mathrm{f}(x)]^{\mathrm{g}(x)}=\left[\lim_{x\to a}\mathrm{f}(x)\right]^{\lim_{x\to a}\mathrm{g}(x)}=l^{\mathrm{m}}\] |
