Special limits

Some limits occur so often in analysis that they have gained special importance. They are used as fundamental reference points in proofs and calculations, especially in connection with trigonometry, exponential functions and logarithms.

 

 

Trigonometric limits

One of the most important limits in all of analysis is:

 

$$ \large \lim_{x \to 0} \frac{\sin x}{x} = 1 $$

 

This limit is used, among other things, to show that the sine function is differentiable at zero, and it forms the basis for the differentiation of trigonometric functions.

 

A closely related limit is:

 

$$ \large \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 $$

 

It is often used together with the first when calculating trigonometric expressions involving both \(\sin x\) and \(\cos x\).

 

 

Exponential and logarithmic functions

Another classic limit defines the number e, which is the base of the natural logarithms:

 

$$ \large \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e $$

 

It shows how a specific repeated percentage growth approaches a constant value as the number of steps becomes very large. This limit forms the basis of the exponential function and its applications in growth models and financial calculations.

 

A related limit is:

 

$$ \large \lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e $$

 

Both forms describe the same relationship between discrete and continuous growth and are often used interchangeably in proofs and derivations.

 

 

Infinite and comparative growth

By comparing functions that grow toward infinity, one can determine which grows faster. For example:

 

$$ \large \lim_{x \to \infty} \frac{\ln x}{x} = 0 $$

$$ \large \lim_{x \to \infty} \frac{x}{e^x} = 0 $$

 

Here, it is seen that the logarithm grows much more slowly than a polynomial, and that even a large polynomial eventually becomes infinitely smaller than an exponential function. These comparisons are often used when analyzing growth rates or determining asymptotic behavior.

 

 

Importance in analysis

Special limits serve as fundamental tools in many proofs and derivations. They appear in the definition of the derivative, in Taylor series, in limits of sequences and in descriptions of growth and convergence. Knowledge of these limits makes it possible to understand and calculate more complex expressions in analysis.