Explanation
Solution
1. Evaluation of Limit:
The given limit is of the form 00. We solve it by rationalizing the numerator twice.
Step 1: First Rationalization
Multiply and divide by the conjugate (1+1+y4+2):
y→0limy4(1+1+y4+2)(1+1+y4−2)(1+1+y4+2)
y→0limy4(1+1+y4+2)(1+1+y4)−2
y→0limy4(1+1+y4+2)1+y4−1
Step 2: Second Rationalization
Now rationalize the term (1+y4−1) by multiplying and dividing by (1+y4+1):
y→0limy4(1+1+y4+2)(1+y4+1)(1+y4−1)(1+y4+1)
y→0limy4(1+1+y4+2)(1+y4+1)(1+y4)−1
y→0limy4(1+1+y4+2)(1+y4+1)y4
Step 3: Simplify and Apply Limit
Cancel y4 and substitute y=0:
2. Analysis of Statements:
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(A) Limit exists and equals 421 → True
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(B) Limit does not exist → False
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(C) Limit exists and equals 221 → False
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(D) Limit exists and equals 22(2+1)1 → False
Identifying False Statements:
Statements B, C, and D are false.
Correct Option: (b)
