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(a) Value

(b) Absolute value

(c) Limit

(d) None of these

Answer

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Hint: General definition of the limit is very important. Whenever learning anything we need to learn the terms associated with it.

Complete step-by-step answer:

Considering the given data, equation can be formed as

$\underset{x\to a}{\mathop{\lim }}\,f(x)=y$

In this we can say that the limit of the function f(x) where the variable $'x'$ tends to $'a'$ which is a constant value.

That is, the value of function f(x) becomes $'y'$ when the variable $'x'$ tends to $'a'$.

It can be also said that:

Function f(x) is defined not only at a particular point but in the neighborhood of the point $'a'$ also.

So, if we substitute the value $'a'$ in the function f(x), we get the value of the function f(x) as $'x'$ tends to $'a'$.

So, $a$is the limit here, f(x) is the function, $y$ is the value of the function at point $'a'$.

Therefore, as $x\to a,f(x)\to l,$i.e., as$'x'$ tends to $'a'$, the function $f(x)$ tends to $'l'$.

So, $l$ is called the value of the function $f(x)$

Hence the correct answer is option (a).

Note: Students often mistake between value and absolute value. Absolute value is the value regardless of the sign. But in this case we cannot ignore the sign. So it is value and not absolute value.

This question shows the importance of theory.

Complete step-by-step answer:

Considering the given data, equation can be formed as

$\underset{x\to a}{\mathop{\lim }}\,f(x)=y$

In this we can say that the limit of the function f(x) where the variable $'x'$ tends to $'a'$ which is a constant value.

That is, the value of function f(x) becomes $'y'$ when the variable $'x'$ tends to $'a'$.

It can be also said that:

Function f(x) is defined not only at a particular point but in the neighborhood of the point $'a'$ also.

So, if we substitute the value $'a'$ in the function f(x), we get the value of the function f(x) as $'x'$ tends to $'a'$.

So, $a$is the limit here, f(x) is the function, $y$ is the value of the function at point $'a'$.

Therefore, as $x\to a,f(x)\to l,$i.e., as$'x'$ tends to $'a'$, the function $f(x)$ tends to $'l'$.

So, $l$ is called the value of the function $f(x)$

Hence the correct answer is option (a).

Note: Students often mistake between value and absolute value. Absolute value is the value regardless of the sign. But in this case we cannot ignore the sign. So it is value and not absolute value.

This question shows the importance of theory.