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  • Evaluate the limit <img alt="\lim_{x\rightarrow \frac{\pi}{2}}f\left ( g\left ( h\left ( x \right ) \right ) \right )" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%

Evaluate the limit \lim_{x\rightarrow \frac{\pi}{2}}f\left ( g\left ( h\left ( x \right ) \right ) \right ) where h\left ( x \right )= \cos x,g\left ( x \right )= x^{3}.

Option: 1

f\left ( 1 \right )


Option: 2

f\left ( 0\right )


Option: 3

f\left ( \frac{\pi}{2} \right )


Option: 4

h\left ( 1 \right )


Answers (1)

best_answer

The limit stated here is \lim_{x\rightarrow \frac{\pi}{2}}f\left ( g\left ( h\left ( x \right ) \right ) \right )\quad \dots\left ( i \right )

where the following data is provided

h\left ( x \right )= \cos x \quad \dots\left ( ii \right )
g\left ( x \right )= x^{3} \quad \dots\left (i ii \right )

Note the following essential points.
 

  • The “Composition law of limit” states that the limit \lim_{x\rightarrow a}f\left ( g\left ( x \right ) \right )= f\left ( \lim_{x\rightarrow a}g\left ( x \right ) \right )= f\left ( A \right ) hold good, if and only if f\left ( x \right )  is continuous at g\left ( x \right )= A.
     
  • The functions \sin \left ( x \right ) and \cos \left ( x \right ) are both continuous for \forall x \in R.


Use the equations (ii) and (iii), and apply the above laws of limit to rewrite the equation (i) in the following way.

\lim_{x\rightarrow \frac{\pi}{2}}f\left ( g\left ( h\left ( x \right ) \right ) \right )
=f\left(\lim _{x \rightarrow-\frac{\pi}{2}} g(h(x))\right)
=f\left(g\left(\lim _{x \rightarrow-\frac{\pi}{2}} h(x)\right)\right)
=f\left(g\left(\lim _{x \rightarrow-\frac{\pi}{2}} \cos x\right)\right)
=f\left(g\left(\cos \left(-\frac{\pi}{2}\right)\right)\right)
=f\left(g\left(\cos \left(\frac{\pi}{2}\right)\right)\right)
= f\left ( g\left ( 0 \right ) \right )
= f\left ( 0^{3} \right )
= f\left ( 0 \right )

Posted by

Gaurav

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