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Evaluate: [("lim")(xvec0)(tan^(-1)x)/x],...

Evaluate: `[("lim")_(xvec0)(tan^(-1)x)/x],` where [.]represent the greatest integer function

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See the graphs of y=x and `tan^(-1)x` in the following figure.

For `f(x)=tan^(-1)(x)=1/(1+x^(2))`
`:." "f'(0)=1`
So the curve touches the line y = x at x = 0
For `xgt0,f'(x)lt1`
So the graph of `y=tan^(-1)x` lies below the graph of y = x.
From the graph, when `xto o^(+)`, the graph of y = x is above the graph of `y = tan^(-1)x`
`or" "tan^(-1)xltxrArr(tan^(-1)x)/xlt1rArr[underset(xto0^(+))("lim")(tanx)/x]=0`
`"Thus "[underset(xto0^(+))("lim")(tanx)/x]=0`
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CENGAGE PUBLICATION-GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS-Exercises
  1. Evaluate: [("lim")(xvec0)(tan^(-1)x)/x], where [.]represent the great...

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  2. Draw the graph of y=tan^(-1)x-cot^(-1)x.

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  3. Find the number of real solutions to the equation 3cos^(-1)x-pix-pi/2=...

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  4. Evalute [lim(xto0) (sin^(-1)x)/(x)]=1, where[*] represets the greatest...

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  5. Solve tan^(-1)xgtcot^(-1)x graphically. Also find where f(x) = max. [t...

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  6. Match the colums.

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  7. Draw the graph of y=cos^(-1)sqrt(log([x])(|x|/x)) where [*] represents...

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  8. Find the value of int(0)^(100pi)sin^(-1)(sin x)dx.

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  9. Draw the graph of y=sin^(-1)((1)/(x))

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  10. Draw the graph of y=sin^(-1)(x^(2))

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  11. Draw the graph of y=cos^(-1)x^(3)" and compare with the graph of "y=co...

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  12. Draw the graph of y=cos^(-1)(2^(x)).

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  13. Draw the graph of y=cos^(-1){x}," where "{*}" represents the fractiona...

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  14. If cos^(-1)(cos x)=(n-x)/n,xge0, has seven roots, then find values of ...

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  15. Draw the graph of f(x)=[cot^(-1)x]," where "[*] represents the greate...

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  16. Draw the graph of y="cosec"("cosec"^(-1)x)" or "y=sec(sec^(-1)x).

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  17. Draw the graph of f(x) =cot^(-1)((2-|x|)/(2+|x|)).

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  18. Draw the graph of y=sin^(-1)(2xsqrt(1-x^(2)))

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  19. Draw the graph of y=(3x-x^(3))/(1-3x^(2)) and hence the graph of y=tan...

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