Discuss the continuity of f(x) at x=0
F(x)= x4+2x3+x2/tan-1x , x(not equal to)0
= 0 , x=0
Answers
we have to discuss the continuity of f(x) at x = 0.
F(x) = (x⁴ + 2x³ + x²)/tan-¹x , x ≠ 0
= 0 , x = 0
taking limit(x→0) F(x)
let's check the form of limit. putting x = 0 in F(x)
we get, 0/0. hence given expression is in the form of limit.
and we can apply L - HOSPITAL RULE
lim(x→0) (4x³ + 6x² + 2x)/(1/1 + x²)
= lim(x→0) (4x³ + 6x² + 2x)(1 + x²)/1
now putting x = 0 we get,
(4 × 0³ + 6 × 0² + 2 × 0)/1 = 0/1 = 0
here we get, lim(x→0) F(x) = 0 and at x = 0, F(x) = 0
so, F(x) is continuous at x = 0.
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