Question

Find the limit............​

Answer 1

Answer:
`-(9)/(x^2)`

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Work Shown:

`\displaystyle \lim_(\Delta x \to 0^(+)) ((9)/(x+\Delta x) - (9)/(x))/(\Delta x)\\\\\\\displaystyle \lim_(\Delta x \to 0^(+)) ((9x)/(x(x+\Delta x)) - (9(x+\Delta x))/(x(x+\Delta x)))/(\Delta x)\\\\\\\displaystyle \lim_(\Delta x \to 0^(+)) ((9x-9(x+\Delta x))/(x(x+\Delta x)))/(\Delta x)\\\\\\\displaystyle \lim_(\Delta x \to 0^(+)) (9x-9x-9\Delta x)/(x\Delta x(x+\Delta x))\\\\\\`

`\displaystyle \lim_(\Delta x \to 0^(+)) (-9\Delta x)/(x\Delta x(x+\Delta x))\\\\\\\displaystyle \lim_(\Delta x \to 0^(+)) (-9)/(x(x+\Delta x))\\\\\\\displaystyle (-9)/(x(x+0))\\\\\\\displaystyle -(9)/(x^2)\\\\\\`

The trick here is to first simplify the expression so that the
`\Delta x` term in the very lower denominator cancels out. This is to avoid dividing by zero. Once this division by zero error is avoided, we can replace the delta x term with 0 and evaluate the limit as
`\Delta x` approaches 0 from the right or positive direction.