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UPSC NDA I 2021 Mathematics Solved Paper

Question : 94 of 120

Marks: +1, -0

What is

\lim\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2}

equal to?

Solution:

Concept:

The limit can be found by direct substitution because the function is continuous at

x = 2

Explanation:

We are given

\displaystyle \lim_{x \to 2} \frac{x^3 + x^2}{x^2 + 3x + 2}

Since the denominator does not become zero at

x = 2

, we can simply substitute

x = 2

directly.

Substitute

x = 2

into the numerator:

2^3 + 2^2 = 8 + 4 = 12

Substitute

x = 2

into the denominator:

2^2 + 3 \cdot 2 + 2 = 4 + 6 + 2 = 12

Thus, the limit equals

\frac{12}{12} = 1

Therefore, the value of the limit is 1.

Answer:

1

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