Question

[tex]f(x) = \begin{cases}
0 & \text{if x is rational} \\
1 & \text{if x is irrational}
\end{cases}[/tex]
​

Answer 1

I think your question is incomplete.

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Method -1: Using formula

$\orange{\boxed{\boxed{\begin{array}{cc}\bf \: \to \:given : \\ \\ \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4 }{x - 2} \\ \\ \rm =\displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - {2}^{2} }{x - 2} \\ \\ \pink{{\boxed{\begin{array}{cc}\bf\;\to \:we \: kmow \: that : \\ \\\displaystyle \lim_{ \rm \: x \to \: a} \: \rm \: \frac{ {x}^{2} - {a}^{2} }{x - a} = n {a}^{n - 1} \end{array}}}} \\ \sf \: apply \: this \: rule \\ \\ = 2 \times {2}^{2 - 1} \\ \\ = 2 \times 2 \\ \\ = 4 \\ \\ \\ \blue{ \boxed{ \therefore \: \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4}{x - 2} = 4}}\end{array}}}}$

Method -2: Simplification the expression

${\boxed{\boxed{\begin{array}{cc}\bf\to \: given : \\ \\ \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4 }{x - 2} \\ \\ \sf \: by \: applying \: limit \: we \: get \: \frac{0}{0} \: form \\ \sf \: which \: is \: undefined. \\ \\ \red{\bf \: so \: simplify \: the \: limit} \\ \\ \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - {2}^{2} }{x - 2} \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{(x + 2) \cancel{(x - 2)}}{ \cancel{(x - 2)}} \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \:(x + 2) \\ \\ \sf \: apply \: limit \\ \\ = 2 + 2 \\ \\ = 4 \\ \\ \blue{ \boxed{ \therefore \: \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4 }{x - 2} = 4}} \end{array}}}}$

Method -3: La'Hospital rules

Rule:

L'Hospital's Rule tells us that if we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.

Now,

$\pink{\boxed{\boxed{\begin{array}{cc} \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4}{x - 2} \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ \frac{d}{dx}( {x}^{2} - 4)}{ \frac{d}{dx}(x - 2) } \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ \frac{d}{dx} \: {x}^{2} - \frac{d}{dx}4 }{ \frac{d}{dx} \: x - \frac{d}{dx} 2} \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{2x - 0}{1 - 0} \\ \\ = \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \:2x \\ \\ = 2 \times 2 \\ \\ = 4 \\ \\ \\ \blue{ \boxed{ \therefore \: \displaystyle \lim_{ \rm \: x \to \: 2} \: \rm \: \frac{ {x}^{2} - 4 }{x - 2} = 4}}\end{array}}}}$