# Limits of Functions

• Functions: $$f\left( x \right),$$ $$g\left( x \right)$$
Argument (independent variable): $$x$$
Real constant numbers: $$a,$$ $$k,$$ $$L,$$ $$\varepsilon,$$ $$\delta$$
1. A function $$f\left( x \right)$$ has the limit $$L$$ as $$x$$ approaches $$a$$ if, for every positive number $$\varepsilon$$, there exists a positive number $$\delta$$ such that $$\left| {f\left( x \right) – L} \right| \lt \varepsilon$$ whenever $$0 \lt \left| {x – a} \right| \lt \delta$$. The limit of a function is denoted by
$$\lim\limits_{x \to \infty } f\left( x \right) = L$$.
2. ### Properties of limits

3. The limit of a constant is equal to the constant:
$$\lim\limits_{x \to a} C = C$$
4. The limit of the sum of two functions is equal to the sum of their limits (assuming these limits exist – this remark applies also to other formulas below):
$$\lim\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] =$$ $$\lim\limits_{x \to a} f\left( x \right) \,+$$ $$\lim\limits_{x \to a} g\left( x \right)$$
5. The limit of the difference of two functions is the difference of their limits:
$$\lim\limits_{x \to a} \left[ {f\left( x \right) – g\left( x \right)} \right] =$$ $$\lim\limits_{x \to a} f\left( x \right) \,-$$ $$\lim\limits_{x \to a} g\left( x \right)$$
6. The limit of the product of two functions is equal to the product of their limits:
$$\lim\limits_{x \to a} \left[ {f\left( x \right) \cdot g\left( x \right)} \right] =$$ $$\lim\limits_{x \to a} f\left( x \right) \cdot \lim\limits_{x \to a} g\left( x \right)$$
7. The limit of the quotient of two functions is the quotient of their limits, provided the limit in the denominator is non-zero:
$$\lim\limits_{x \to a} {\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} =$$ $${\large\frac{{\lim\limits_{x \to a} f\left( x \right)}}{{\lim\limits_{x \to a} g\left( x \right)}}\normalsize},$$ if $$\lim\limits_{x \to a} g\left( x \right) \ne 0.$$
8. A constant factor can be taken out of the limit:
$$\lim\limits_{x \to a} \left[ {kf\left( x \right)} \right] =$$ $$k\lim\limits_{x \to a} f\left( x \right)$$
9. Limit of a composite function
$$\lim\limits_{x \to a} f\left( {g\left( x \right)} \right) =$$ $$f\left( {\lim\limits_{x \to a} g\left( x \right)} \right)$$
10. Limit of a continuous function
If the function $$f{\left( x \right)}$$ is continuous at $$x=a,$$ then
$$\lim\limits_{x \to a} f\left( x \right) = f\left( a \right)$$.
11. ### Some special limits

12. $$\lim\limits_{x \to 0} {\large\frac{{\sin x}}{x}\normalsize} = 1$$
13. $$\lim\limits_{x \to 0} {\large\frac{{\tan x}}{x}\normalsize} = 1$$
14. $$\lim\limits_{x \to 0} {\large\frac{{\arcsin x}}{x}\normalsize} = 1$$
15. $$\lim\limits_{x \to 0} {\large\frac{{\arctan x}}{x}\normalsize} = 1$$
16. $$\lim\limits_{x \to 0} {\large\frac{{\ln\left( {1 + x} \right)}}{x}\normalsize} = 1$$
17. $$\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{1}{x}}\normalsize} \right)^x} = e$$
18. $$\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{k}{x}}\normalsize} \right)^x} = e^k$$
19. $$\lim\limits_{x \to 0} {a^x} = 1$$