Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to {3^ - }} \frac{{\sqrt x }}{{{{\left( {x - 3} \right)}^5}}}\)

Short answer

The limit tends to negative infinity.

Step-by-step solution

Analyze the given function for any discontinuity

Consider the limit\(\mathop {\lim }\limits_{x \to {3^ - }} \frac{{\sqrt x }}{{{{\left( {x - 3} \right)}^5}}}\).

The given function’s denominator hasnumeratortending to\(\sqrt 3 \)anddenominatortending to 0 from leftas\(x\)approaches 3 from the left side.

Since the denominator tends to zero from left as\(x\)approaches from the left of 3, thefunction becomesdiscontinuous as \(x \to {3^ - }\).

Estimate the limit

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

\(\begin{aligned}\mathop {\lim }\limits_{x \to {3^ - }} \frac{{\sqrt x }}{{{{\left( {x - 3} \right)}^5}}} &= \frac{{\sqrt 3 }}{{{0^ - }}}\\ &= - \infty \end{aligned}\)

Thus, the limit tends to negative infinity.