Operations with limits

We already know the function limit. In this topic, we will learn what operations we can perform with it.

Operations with limits

Suppose we have two functions $f(x)$ and $g(x)$, and two limits $\lim\limits_{x \to a} f(x)$ and $\lim\limits_{x \to a} g(x)$

Then we can perform the following operations with limits:

1) Add

$$\lim\limits_{x \to a}f(x) + \lim\limits_{x \to a}g(x)=\lim\limits_{x \to a}( f(x) + g(x) )$$2) Subtract

$$\lim\limits_{x \to a}f(x) -\lim\limits_{x \to a}g(x)=\lim\limits_{x \to a}( f(x) - g(x) )$$3) Multiply

$$\lim\limits_{x \to a}f(x) \cdot\lim\limits_{x \to a}g(x)=\lim\limits_{x \to a}( f(x) \cdot g(x) )$$4) If $\lim\limits_{x \to a} g(x) \ne 0$, we can divide $\lim\limits_{x \to a} f(x)$ by $\lim\limits_{x \to a} g(x)$

$$\dfrac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}=\lim\limits_{x \to a} \dfrac{f(x)}{g(x)}$$5) Multiply by a constant $c$$$c \cdot \lim\limits_{x \to a}f(x)=\lim\limits_{x \to a} (c \cdot f(x))$$6) Raise to a positive integer power $n$

$(\lim\limits_{x \to a} f(x))^n=\lim\limits_{x \to a} (f(x))^n$

7) Extract $n$th root, where $n$ is a positive integer

$\sqrt[n]{\lim\limits_{x \to a}f(x)}=\lim\limits_{x \to a} \sqrt[n]{f(x)}$

Example

Let's do some practice and find the limit $$\lim\limits_{x \to 1} \dfrac{x^4 -3x^2+1}{4-x}$$We use the rules that we've considered above:

$$\lim\limits_{x \to 1} \dfrac{x^4 -3x^2+1}{4-x} = \dfrac{\lim\limits_{x \to 1} (x^4 -3x^2+1)}{\lim\limits_{x \to 1} (4-x)} = \dfrac{\lim\limits_{x \to 1} x^4 -3\lim\limits_{x \to 1} x^2 + \lim\limits_{x \to 1} 1}{\lim\limits_{x \to 1} 4-\lim\limits_{x \to 1} x} = \\ \ \\ =\dfrac{1^4-3 \cdot 1^2+1}{4-1} = - \dfrac{1}{3}$$

Conclusion

In this topic, we learned about operations with limits. If the limits of functions $f(x)$ and $g(x)$ both exist, then we can add these limits, subtract, multiply and divide (provided that the denominator is not $0$), multiply by a constant, or raise to a power. These operations are very helpful in calculating all sorts of limits.