Q.46

The limit of the function lim

𝑥→2

( 2𝑥 2 𝑥 + 2 2 − 𝑥 4 −12) is _____________. (rounded off to 1

decimal)

Limit of a Function – Solved Example (Rounded to 1 Decimal)

Question (Q.46)

Evaluate the limit of the function:

lim_x→2

(2x² + 2x − 12) / (x² − 4)

(Round off the answer to 1 decimal place)

Final Answer

2.5

Step-by-Step Solution

Step 1: Substitute x = 2

Numerator:

2(2)² + 2(2) − 12 = 8 + 4 − 12 = 0

Denominator:

(2)² − 4 = 4 − 4 = 0

Since the result is 0/0, it is an indeterminate form and must be simplified.

Step 2: Factorize Numerator and Denominator

Numerator:

2x² + 2x − 12 = 2(x² + x − 6)
= 2(x + 3)(x − 2)

Denominator:

x² − 4 = (x + 2)(x − 2)

Step 3: Cancel Common Terms

(2(x + 3)(x − 2)) / ((x + 2)(x − 2))

Cancel (x − 2):

= 2(x + 3) / (x + 2)

Step 4: Substitute x = 2

2(2 + 3) / (2 + 2) = 10 / 4 = 2.5

Rounded to 1 decimal place: 2.5

Explanation of Possible Options

Key Concepts Used

• Limits of a function
• Indeterminate form (0/0)
• Factorization
• Algebraic simplification
• Rounding off

Conclusion

This problem demonstrates how factorization helps resolve indeterminate forms in limits.
Such questions are very common in BCA, BSc, and engineering entrance examinations.