Functions and limits - Standard limits
- Standard limits are specific values that help to determine the limits of functions as they approach certain values.
- There are six standard limits, including the limit of a constant function, the limit of a linear function, the limit of a quadratic function, the limit of a square root function, the limit of a rational function, and the limit of an exponential function.
- Knowing the standard limits can help you quickly determine the limit of more complex functions by applying them to smaller parts of the larger function.
- The limit of a function is the value that the function approaches as the input approaches a certain value, and may or may not be equal to the actual value of the function at that point.
- Understanding standard limits is a crucial part of calculus, as they form the foundation for more advanced topics such as derivatives and integrals.
Important Formulas to Remember in these Functions and limits
- $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
- $$\lim_{x \to 0} \frac{\tan x}{x} = 1$$
- $$\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$$
- $$\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$$
- $$\lim_{x \to \infty} \left(1+\frac{k}{x}\right)^x = e^k$$
- $$\lim_{x \to 0} \frac{e^x-1}{x} = 1$$
- $$\lim_{x \to \infty} \frac{\ln x}{x} = 0$$
- $$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$$
- $$\lim_{x \to \infty} \frac{a^x}{x!} = 0$$
- $$\lim_{x \to \infty} \frac{x^k}{a^x} = 0$$
- $$\lim_{x \to \infty} \sqrt[x]{x} = 1$$