Limits Law of Addition of functions

Formula

$\large \displaystyle \lim_{x \to a} \, [f(x) + g(x)]$ $\large \,=\,$ $\large \displaystyle \lim_{x \to a} f(x)$ $\large +$ $\large \displaystyle \lim_{x \to a} g(x)$

Proof

$f(x) = x^2$ and $g(x) = 2x$ are two functions.

Step: 1

The limit of $f(x)$ and $g(x)$ functions when the value of $x$ approaches to $a$ are written as follows.

$\displaystyle \lim_{x \to a} \, f(x) = \displaystyle \lim_{x \to a} \, x^2$

$\implies \displaystyle \lim_{x \to a} \, x^2 = a^2$

$\displaystyle \lim_{x \to a} \, g(x) = 2x$

$\implies \displaystyle \lim_{x \to a} \, 2x = 2a$

Step: 2

Add the values of both functions.

$\displaystyle \lim_{x \to a} \, f(x) + \displaystyle \lim_{x \to a} \, g(x) = a^2 + 2a$

Step: 3

Now, add both functions and find the value of sum of them when the limit $x$ is tending to $a$.

$\displaystyle \lim_{x \to a} \, [f(x) + g(x)] = \displaystyle \lim_{x \to a} \, (x^2 + 2x)$

$\implies \displaystyle \lim_{x \to a} \, (x^2 + 2x) = a^2 + 2a$

$\therefore \,\,\,\,\,\, \displaystyle \lim_{x \to a} \, [f(x) + g(x)] = a^2 + 2a$

Step: 4

Observe the results of second and third steps.

$\displaystyle \lim_{x \to a} \, f(x) + \displaystyle \lim_{x \to a} \, g(x) = a^2 + 2a$

$\displaystyle \lim_{x \to a} \, [f(x) + g(x)] = a^2 + 2a$

$\therefore \,\,\,\,\,\, \displaystyle \lim_{x \to a} \, [f(x) + g(x)]$ $=$ $\displaystyle \lim_{x \to a} \, f(x) + \displaystyle \lim_{x \to a} \, g(x)$

Therefore, the property of the limit of sum two functions is derived in mathematics from an algebraic example.

The fundamental limits law is not limited to two functions and the addition limits identity can also be applied to more than two functions.

$\therefore \,\,\,\,\,\, \displaystyle \lim_{x \to a} \, [f(x) + g(x) + h(x) + \cdots]$ $=$ $\displaystyle \lim_{x \to a} \, f(x)$ $+$ $\displaystyle \lim_{x \to a} \, g(x)$ $+$ $\displaystyle \lim_{x \to a} \, h(x) + \cdots$

Verification

$\sin x +\cos x$ is an example function and find the value of this function when $x$ tends to $\dfrac{\pi}{4}$ to verify the limits rule.

Step: 1

$\displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, (\sin x + \cos x)$ $=$ $\sin \Big(\dfrac{\pi}{4}\Big) + \cos \Big(\dfrac{\pi}{4}\Big)$

$\implies \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, (\sin x + \cos x)$ $=$ $\dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}$

$\implies \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, (\sin x + \cos x)$ $=$ $\dfrac{2}{\sqrt{2}}$

$\therefore \,\,\,\,\,\, \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, (\sin x + \cos x)$ $=$ $\sqrt{2}$

Step: 2

Now, calculate the value of limit of each function in the expression when the value of $x$ is approaching $\dfrac{\pi}{4}$.

$\displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x$ $=$ $\sin \Big(\dfrac{\pi}{4}\Big)$

$\implies \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x$ $=$ $\dfrac{1}{\sqrt{2}}$

$\displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\cos \Big(\dfrac{\pi}{4}\Big)$

$\implies \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\dfrac{1}{\sqrt{2}}$

Now find the value of both functions by the addition.

$\displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x + \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}$

$\implies \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x + \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\dfrac{2}{\sqrt{2}}$

$\therefore \,\,\,\,\,\, \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x + \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\sqrt{2}$

Step: 3

The limit of sum of two or more functions is equal to the summation of the limit of each function.

$\therefore \,\,\,\,\,\, \displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, (\sin x + \cos x)$ $=$ $\displaystyle \lim_{\displaystyle x \to \frac{\pi}{4}} \, \sin x + \lim_{\displaystyle x \to \frac{\pi}{4}} \, \cos x$ $=$ $\sqrt{2}$

Thus, the addition identity of limits has been verified by an example in mathematics.

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