# 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 two words are used to represent closeness. The right arrow ($\rightarrow$) symbol is used to denote the closeness in mathematics.

## 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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