1> \lim_{x\rightarrow 0}\frac{e^{-4x}-1}{e^{-2x}+e^{-x}-2}
2>\lim_{x\rightarrow -1^{+}}\frac{\sqrt{\pi}-\sqrt{cos^{-1}x}}{\sqrt{x+1}}
3> \lim_{x\rightarrow 1}(1-x)tan(\frac {\pi x}{2})
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2 Answers
11Ans 1)Simply Use L'H Rule ,
L = 4
Ans 2) Put y = cos -1 x, so that x = cos y
Therefore, if x → -1, that implies y→ ∩
Therefore, L = lim x→∩+ √∩ - √y√(1 + cos y)
= lim x→ ∩+ √∩ - √y√2 cos (y/2)
Since it is indeterminate form, use L' H Rule
Ans 3)L = lim x→1 (1-x) tan (∩x/2)
= lim x→1 (1-x) cot [∩/2 (1-x)]
= lim x→1 1-xSin [∩/2 (1-x)] . cos [∩/2 (1-x)]
Put [∩/2 (1-x)] = y
L = lim y→0 ysin y . cos y = 1