Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to 1} \,\frac{{{x^2} + 2x}}{{{x^2} - 2x + 1}}\)

Short answer

The limit tends to positive infinity.

Step-by-step solution

Analyze the given function at the given point

The given function’sdenominator is a whole square of\({\left( {x - 1} \right)^2}\)as\({x^2} - 2x + 1 = {\left( {x - 1} \right)^2}\),which is tending to 0 and hasnumerator tending to 3, as\(x\)approaches 1.

Since the denominator tends to zero as \(x\) approaches 1, so the function becomes discontinuous as \(x \to 1\).

Estimate the limit

The numerator tends to a positive value, and the denominator of the function tends to 0 as\(x\)approaches 1.

Thus, the final limit must approach to infinity as shown below:

\(\begin{aligned}\mathop {\lim }\limits_{x \to 1} \frac{{{x^2} + 2x}}{{{x^2} - 2x + 1}} &= \frac{3}{0}\\ &= \infty \end{aligned}\)

Thus the limit tends to infinity.