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				   please explain the wavy curve method and la hospital rule with examples


6 years ago

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										Dear Abhik

Wavy curve method :--

1 . put only odd power factors in numerator and denominator  equal to zero separately as function changes sign only for odd power
2.  Plot these points  on the number line in increasing order  .
3. Now check the coefficients of X and make them positive.
4. expression will be +ve to the right of the highest point allotted on number line.
5.  alternately assign + and - signs in rest of intervals ( from right to left )

to see examples see the pic

L'Hôpital's rule
l'Hôpital's rule states that for functions ƒ and g:
If $\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0 \,$ or $\pm\infty$ and $\lim_{x\to c}f'(x)/g'(x)$ exists,
then $\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}.$
Here is  example involving 0/0:
$\lim_{x\to 0}{\frac{e^x-1-x}{x^2}} =\lim_{x\to 0}{\frac{e^x-1}{2x}} =\lim_{x\to 0}{\frac{e^x}{2}}={\frac{1}{2}}.$
Here is another example involving ∞/∞:

$\lim_{x\to 0^+} x \ln x =\lim_{x\to 0^+}{\frac{\ln x}{1/x}} =\lim_{x\to 0^+}{\frac{1/x}{-1/x^2}} =\lim_{x\to 0^+} -x = 0.$

All the best.
AKASH GOYAL

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6 years ago

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