Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } - 1 }$$

Short answer

The limit of the given function as \(x\) approaches 1 is \(0.5\).

Step-by-step solution

Graphical Method

The first approach should be to plot the function \(f(x) = \frac { x - 1 } { x ^ { 2 } - 1 }\). As \(x\) approaches 1, observe the direction and value the function is approaching. That will provide an intuitive understanding of the limit.

Algebraic Method - Step 1: Identifying the form

There would be a direct substitution for \(x = 1\) in the function. \(f(1) = \frac { 1 - 1 } { 1 ^ { 2 } - 1 } = \frac { 0 } { 0 }\). This is an indeterminate form, so the limit isn't immediately apparent and requires further manipulation.

Algebraic Method - Step 2: Factorisation

Given the equation \(\frac { x - 1 } { x ^ { 2 } - 1 }\), the denominator \(x^{2} - 1\) can be factored to \((x-1)(x+1)\) via the difference of two squares identity. Hence, the function becomes \(\frac { x - 1 } { (x-1)(x+1) }\).

Algebraic Method - Step 3: Simplification

Cancel out the common factors in the numerator and denominator, the function simplifies to \(\frac {1}{x+1}\). Now, resubstitute \(x = 1\) to get the limit.

Algebraic Method - Step 4: Substitution of Limit Point

Now subtitute \(x = 1\) into the simplified function \( \frac {1}{ x + 1 }\), hence the limit as \(x\) approaches \(1\) is \(\frac{1}{1+1} = \frac{1}{2}\). Therefore, the limit of the function as \(x\) approaches 1 is \(0.5\).