Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to 2} \frac{{{x^2}}}{{{{\left( {x - 2} \right)}^2}}}\)

Short answer

The limit tends to infinity.

Step-by-step solution

Analyze the given function for any discontinuity

Consider the limit\(\mathop {\lim }\limits_{x \to 2} \frac{{{x^2}}}{{{{\left( {x - 2} \right)}^2}}}\).

The given function’sdenominator has anumerator tending to 4 and a numerator tending to 0 as\(x\)approaches 2.

Since the denominator tends to zero as\(x\)approaches 2, so thefunctionbecomesdiscontinuousat \(x = 2\).

Estimate the limit

Plugin the values of denominator and numerator into the function and simplify to get the required limit as:

\(\begin{aligned}\mathop {\lim }\limits_{x \to 2} \frac{{{x^2}}}{{{{\left( {x - 2} \right)}^2}}} &= \frac{4}{0}\\ &= \infty \end{aligned}\)

Thus, \(\mathop {\lim }\limits_{x \to 2} \frac{{{x^2}}}{{{{\left( {x - 2} \right)}^2}}} = \infty \).