Question

How do you evaluate the limit \[\dfrac{x-1}{{{x}^{2}}-1}\] as \[x\] approaches 1?

Answer 1

Hint: The above given phrase can be written in a mathematical form as \[\displaystyle \lim_{x \to 1}\dfrac{x-1}{{{x}^{2}}-1}\]. We can see in the denominator that the equation is \[{{x}^{2}}-1\]. We can expand this expression using the formula \[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b)\]. We will now have the denominator as \[(x-1)(x+1)\]. We can see that the numerator and denominator has similar terms so it will get cancelled and then we will apply \[x \to 1\]. Hence, we will get the value of the limit.

Complete step by step solution:
According to the given question, we have to solve the equation for \[x\] approaching 1. Firstly, we will begin by writing the mathematical form of the given question, we have,
\[\displaystyle \lim_{x \to 1}\dfrac{x-1}{{{x}^{2}}-1}\]-----(1)
We cannot straightaway apply the value of \[x\] as 1, as doing so will cause the expression to have the form \[\dfrac{0}{0}\], which is an indeterminate form.
So, we will modify the given expression and after then only we will put \[x\] as 1.
In equation (1), we can see that the denominator has an equation which is \[{{x}^{2}}-1\] and this can be expanded using an algebraic formula and that is, \[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b)\]
Applying this formula in the denominator, we have the equation (1) as,
\[\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{(x-1)}{(x-1)(x+1)}\]
We now can see that both the numerator and the denominator have a similar term, \[(x-1)\] and so this term can be cancelled and the new expression we have is,
\[\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{1}{(x+1)}\]
Now, we will apply the limits \[x \to 1\] in the above expression and we get,
\[\Rightarrow \dfrac{1}{1+1}\]
We have,
\[\Rightarrow \dfrac{1}{2}\]
Therefore, \[\displaystyle \lim_{x \to 1}\dfrac{x-1}{{{x}^{2}}-1}=\dfrac{1}{2}\].

Note: The expression should be evaluated and reduced in such a way that, when limits are applied to the expression, it should be give the result in the indeterminate forms such as \[\dfrac{0}{0}\] or \[\dfrac{\infty }{\infty }\]. Also, certain basic formulae are to be memorized so that solving the expression gets faster.