Limit

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})

2 Answers

Ans 1)Simply Use L'H Rule ,
L = 4

Ans 2) Put y = cos^-1 x, so that x = cos y
Therefore, if x â†’ -1, that implies yâ†’ âˆ©

Therefore, L = lim xâ†’âˆ©+ âˆšâˆ© - âˆšyâˆš(1 + cos y)

= lim xâ†’ âˆ©+ âˆšâˆ© - âˆšyâˆš2 cos (y/2)

Since it is indeterminate form, use L' H Rule

Ans 3)L = lim xâ†’1 (1-x) tan (âˆ©x/2)
= lim xâ†’1 (1-x) cot [âˆ©/2 (1-x)]

= lim xâ†’1 1-xSin [âˆ©/2 (1-x)] . cos [âˆ©/2 (1-x)]
Put [âˆ©/2 (1-x)] = y
L = lim yâ†’0 ysin y . cos y = 1

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