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Below are few problems based on Limit and derivative:

Example 1: Solve $\lim_{x\rightarrow 2}$ $\frac{2x^{2}+3x -4}{4-5x}$

Solution :

$\frac{\lim_{x\ \rightarrow\ 2}(2x^{2}\ +\ 3x\ -\ 4)}{\lim_{x\ \rightarrow\ 2}(4\ -\ 5x)}$

= $\frac{\lim_{x\rightarrow 2}2x^{2}\ +\ lim_{x\ \rightarrow\ 2}3x\ -\ \lim_{x\ \rightarrow\ 2}\ 4}{\lim_{x\ \rightarrow\ 2} 4\ -\ \lim_{x\ \rightarrow\ 2}\ 5x}$

= $\frac{2(2)^{2}+3(2)\ -\ 4}{4\ -\ 5(2)}$

= $\frac{10}{-\ 6}$

= $- \frac{5}{3}$

Example 2: Find $\frac{dy}{dx}$ for x$^{2}$ + y$^{2}$ - 8xy = 6

Solution : Given : X$^{2}$ + y$^{2}$ - 8xy = 6

Differentiating with respect to x we get

2x + 2y $\frac{dy}{dx}$ - 8y - 8x$\frac{dy}{dx}$ = 0

$\frac{dy}{dx}$ (2y - 8x) = 8y - 2x

$\frac{dy}{dx}$ = $\frac{4y - x}{y -4x}$

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