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. \(\displaystyle\lim_{x\to 0} a^x=1\)

Functions and Their Graphs

Limits of Functions

Definition and Properties of the Derivative

Table of Derivatives

Higher Order Derivatives