Rationalization
There are two scenarios where you might want to consider employing this technique. The first scenario arises when you encounter an indeterminate form, specifically $\frac{0}{0}$, and the denominator takes the form of $\sqrt{\cdot}-\sqrt{\cdot}$. The second scenario occurs when you are working with a limit as $x$ approaches infinity, and direct evaluation leads to the indeterminate form $\infty - \infty.” These are two common situations where the technique we’ll discuss becomes particularly useful.
To illustrate these scenarios, let’s explore two typical examples.
| Example 1. $\displaystyle\lim_{x\to 0}\frac{x^2}{\sqrt{ | x^2+2 | }-\sqrt{ | x^2-2 | }}$ |
Example 2. $\displaystyle\lim_{x\to\infty}\sqrt{x}-\sqrt{x+1}$
An important application of the squeeze theorem
To compute $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{x}$, we rely on the squeeze theorem. It turns out that $\lim_{x\to 0}\frac{\sin(x)}{x}=1$ serves as a tool for evaluating various limits involving the sine function.
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Concepts of infinite
Let’s consider the concept of infinity as a symbol, one that holds a unique place in mathematics—it surpasses any finite number. Embracing this idea, we can establish a precise definition for limits such as $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to \infty}f(x)=L$. In computational terms, this symbol exhibits the following algebraic properties:
- $\frac{1}{\infty} = 0$
- $\infty + \infty = \infty$
- $\infty\cdot\infty = \infty$
Of particular interest is the first property, which leads to a valuable technique for evaluating limits as $x$ approaches infinity by focusing on the dominant term in the expression.
Evaluate $\displaystyle\lim_{x\to\infty}\sqrt{2x^2+1}\sin\left(\frac{1}{x}\right)$.