Correct Answer - Option 2 : 3
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: \(\mathop {\lim }\limits_{x \to 1} \frac{{{x^2} + 3x + 2}}{{{x^2} + 1}} = k\)
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 \(\mathop {\lim }\limits_{x \to 1} \frac{{{x^2} + 3x + 2}}{{{x^2} + 1}}\)does not result into any indeterminate form
So, we can substitute x = 1 in the expression \(\frac{{{x^2} + 3x + 2}}{{{x^2} + 1}}\) in order to find the value of k
⇒ \(\mathop {\lim }\limits_{x \to 1} \frac{{{x^2} + 3x + 2}}{{{x^2} + 1}} = 3 = k\)
Hence, Option B is the correct answer.