Correct Answer - Option 4 : 4
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.