Math Doubts

Approach

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.

approaches or tends to

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.

  1. $x$ approaches $a$.
  2. $x$ tends to $a$.

Example

$x$ 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 $a = 6$.

Case: $1$

Take $x = 5$. The values of $x$ and $a$ are not equal because their quantities are not same ($5 \ne 6$). Therefore, $x \ne a$.

Case: $2$

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.

  1. The value of $5.9$ approaches to the value of $6$.
  2. The value of $5.9$ tends to the value of $6$.

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.

Case: $3$

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.

Note

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.

  1. $y \to 7$. It expresses that the value of $y$ approaches $7$ but not equal to $7$.
  2. $h \to 0$. It expresses that the value of $h$ tends to $0$ but it doesn’t equal to $0$.
  3. $m \to n$. It expresses that the value of $m$ approaches to $n$ but they are not equal.
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