# Differential calculus - Limits of functions

Functions : $f(x),g(x)$
Argument : x
Real constants : a, k
1. $\displaystyle\lim_{x \to a}[f(x)+g(x)] =\lim_{x \to a}f(x)+\lim_{x \to a}g(x)$
2. $\displaystyle\lim_{x \to a}[f(x)-g(x)] =\lim_{x \to a}f(x)-\lim_{x \to a}g(x)$
3. $\displaystyle\lim_{x \to a}[f(x).g(x)] =\lim_{x \to a}f(x).\lim_{x \to a}g(x)$
4. $\displaystyle\lim_{x \to a}{f(x) \over g(x)}={\displaystyle\lim_{x \to a}f(x) \over \displaystyle\lim_{x \to a}g(x) }$, if $\displaystyle\lim_{x\to a} g(x) \neq 0.$
5. $\displaystyle\lim_{x\to a} f[kf(x)]= k\lim_{x\to a}f(x)$
6. $\displaystyle\lim_{x\to a} f(g(x))= f(\lim_{x\to a}g(x))$
7. $\displaystyle\lim_{x\to a} f(x)=f(a),$if the function f(x) is continuous at x=a.
8. $\displaystyle\lim_{x\to 0}{sin \,x\over x} =1$
9. $\displaystyle\lim_{x\to 0}{tan \,x\over x} =1$
10. $\displaystyle\lim_{x\to 0}{-sin \,x\over x} =1$
11. $\displaystyle\lim_{x\to 0}{-tan \,x\over x} =1$
12. $\displaystyle\lim_{x\to 0}{In(1+x)\over x} =1$
13. $\displaystyle\lim_{x\to \infty}\bigg(1+{1 \over x}\bigg )^x=e$
14. $\displaystyle\lim_{x\to \infty}\bigg(1+{k \over x}\bigg )^x=e^k$
15. $\displaystyle\lim_{x\to 0} a^x=1$