CONCEPT:

If \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) does not result into indeterminate form, then we use direct substitution in order to find the limits.

The are 7 indeterminate forms which are as follows:

• \((\frac{0}{0})\)
• \(\left( {\frac{{ \pm ∞ }}{{ \pm ∞ }}} \right)\)
• (∞ - ∞)
• (0 × ∞)
• 00
• 1∞
• ∞0

CALCULATION:

Given: \(\mathop {\lim }\limits_{x \to 0} \frac{{-{3x^2} - 7x + 8}}{{{7x^2} + 2x + 2}} = k\)

As we know that, if \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\)does not result into indeterminate form, then we use direct substitution in order to find the limits.

Here, also we can see that \(\mathop {\lim }\limits_{x \to 0} \frac{{-{3x^2} - 7x + 8}}{{{7x^2} + 2x + 2}}\)does not result into any indeterminate form

So, we can substitute x = 0 in the expression \(\frac{{-{3x^2} - 7x + 8}}{{{7x^2} + 2x + 2}}\) in order to find the value of k

⇒ \(\mathop {\lim }\limits_{x \to 0} \frac{{-{3x^2} - 7x + 8}}{{{7x^2} + 2x + 2}} = 4 = k\)

Hence, Option D is the correct answer.