Question

What are the applications of limits?

Answer 1

Hint: Let us consider $f\left( x \right)$ is a real-valued function and $p$ is a real number, then the expression $\displaystyle \lim_{x \to p}f\left( x \right)=T$ means that $f\left( x \right)$ can be as close to $T$ as desired by making $x$ it sufficiently close to $p$ . In such a case, we say that the limit of $f$ , as $x$ approaches to $p$ , is $T$ .
This statement is true even if $f\left( p \right)\ne T$ . Indeed the function $f\left( x \right)$ need not even be defined at $p$ .
Use this to find the applications in science.

Complete step by step solution:
There is the application of limits in mathematics and day to day life.
The application of limits in mathematics is,
Limits also play an essential role in the calculation of differentiation and integration which are collectively known as The Calculus.
They are also used to decide if a particular function is continuous over the left- and right-hand limits.
Limits are used to find the left- and right-hand limits of a given function.
For example,
Let us now evaluate the left- and right-hand limits of the function $f\left( x \right)=\left\{ \begin{matrix}
   1+{{x}^{2}},if\left( 0\le x<1 \right) \\
   2-x,if\left( x>1 \right) \\
\end{matrix} \right\}$ at $x=1$ . Also, show that $\displaystyle \lim_{x \to 1}f\left( x \right)$ does not exist.
LHL of $f\left( x \right)$ at $x=1$ is,
$\Rightarrow \displaystyle \lim_{x \to {{1}^{-}}}=\displaystyle \lim_{h\to 0}f\left( 1-h \right)$
Since $1+{{x}^{2}},if\left( 0\le x<1 \right)$
$\Rightarrow \displaystyle \lim_{x \to {{1}^{-}}}=\displaystyle \lim_{h\to 0}f\left[ 1+{{\left( 1-h \right)}^{2}} \right]$
$\Rightarrow \displaystyle \lim_{x \to {{1}^{-}}}=\displaystyle \lim_{h\to 0}f\left[ 2-2h+{{h}^{2}} \right]=2$
RHL of $f\left( x \right)$ at $x=1$ is,
$\Rightarrow \displaystyle \lim_{x \to {{1}^{+}}}=\displaystyle \lim_{h\to 0}f\left( 1+h \right)$
Since $2-x,if\left( x>1 \right)$
$\Rightarrow \displaystyle \lim_{x \to {{1}^{-}}}=\displaystyle \lim_{h\to 0}f\left[ 2-\left( 1+h \right) \right]$
$\Rightarrow \displaystyle \lim_{x \to {{1}^{-}}}=\displaystyle \lim_{h\to 0}f\left[ 1-h \right]=1$
Clearly, $\displaystyle \lim_{x \to {{1}^{-}}}f\left( x \right)\ne \displaystyle \lim_{x \to {{1}^{+}}}f\left( x \right)$
So, $\displaystyle \lim_{x \to 1}f\left( x \right)$ does not exist.
Coming to the day-to-day life applications,
It is utilized to quantify the temperature of ice shapes in warm water.
It is also used to measure the strength of electric, magnetic, and gravitational fields.
It is also used to measure the instantaneous speed of the object.

Note: limits are the strategy by which the subsidiary, or pace of progress, of a capacity is determined, and they are utilized all through examination as a method of making approximations into careful amounts, as when the zone inside a curved area is characterized to be the limit of approximations by rectangles.