The words “Approach” and “Tend” are often used in calculus and it’s a basic concept to start learning the limits.
According to English language, the real meaning of “Approach” or “Tend” is, come near or nearer to something.
For example, if a point comes nearly to another point, then it’s said that the point approaches to another point. In calculus, the word “Approach” is symbolically represented by right arrow ($\rightarrow$) symbol.
Mathematical, it is written as $x \,\to\, a$ and it is read as two ways.
$x$ is a variable and $a$ is a constant. The value of $x$ starts from $5$ but less than $6$ and the value of $a$ is $6$. Therefore, $x \in [5, 6)$ and $a = 6$.
Take $x = 5$. The values of $x$ and $a$ are not equal because their quantities are not same ($5 \ne 6$). Therefore, $x \ne a$.
Take $x = 5.9$. The value of $5.9$ is not equal to $6$ but its value is closely equal to $6$. In other words, $5.9 \approx 6$. It is expressed in two ways in calculus.
Mathematically, it is written as $5.9 \,\to\, 6$. Therefore, $x \,\to\, a$. Remember, the right arrow represents that the value of $x$ closes the value of $a$ but they’re not equal.
Now, take $x = 5.99$. In this case also, the value of $5.99$ does not equal to $6$ but approximately equals to $6$. Therefore, $5.99 \approx 6$.
Therefore, $x \,\to \, a$ in this case.
The meaning of $x \,\to\, a$ is, the value of $x$ can be anything, which is close to the value of $a$ but not equal to $a$. So, the value of $x$ can be any value closes to $6$, for example $5.9$, $5.95$, $5.991$, $5.999$ and so on.
Let’s study the concept of approach or tend in calculus with some more examples.
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