Question

question 11
evaluate the limit: \\(\lim_{x\to 2^+} \frac{x + 7}{(x - 2)^2}\\)
\\(\circ \\ \infty\\)
\\(\circ \\ \text{no correct answer choice is given.}\\)
\\(\circ \\ 7/2\\)
\\(\circ \\ -7/2\\)
\\(\circ \\ -\infty\\)

Explanation:

Step1: Analyze numerator and denominator as \( x \to 2^+ \)

As \( x \) approaches \( 2 \) from the right (\( x \to 2^+ \)), the numerator \( x + 7 \) approaches \( 2 + 7 = 9 \) (a positive finite number). The denominator \( (x - 2)^2 \): since \( x > 2 \), \( x - 2 \) is positive, and squaring a positive number gives a positive number approaching \( 0 \) (i.e., \( (x - 2)^2 \to 0^+ \)).

Step2: Determine the limit behavior

A positive finite number divided by a positive number approaching \( 0 \) will tend to \( +\infty \). So \( \lim_{x \to 2^+} \frac{x + 7}{(x - 2)^2} = \infty \).

Answer:

\( \infty \) (corresponding to the option with \( \infty \))