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  • Evaluate the limit <img alt="\lim _{x \rightarrow \frac{\pi}{4}} f\left(\left((g(x))^2-h(x)\right)\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%20%5Cf

Evaluate the limit \lim _{x \rightarrow \frac{\pi}{4}} f\left(\left((g(x))^2-h(x)\right)\right) where h(x)=\sin x-\cos x, g(x)=\sin x+\cos x.

Option: 1

f\left ( 0 \right )


Option: 2

f\left ( 1 \right )


Option: 3

f\left ( \sqrt{2} \right )


Option: 4

f\left ( 2 \right )


Answers (1)

best_answer

The limit stated here is \lim _{x \rightarrow \frac{\pi}{4}} f\left(\left((g(x))^2-h(x)\right)\right)\quad \dots\left ( i \right )

where the following data is provided

h(x)=\sin x-\cos x \quad \dots\left ( ii \right )
g(x)=\sin x+\cos x \quad \dots\left ( iii \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 “Difference law for limits” states that \lim_{x\rightarrow a}f\left ( x \right )-\lim_{x\rightarrow a}g\left ( x \right )= \lim_{x\rightarrow a}\left [ f\left ( x \right )+g\left ( x \right )\right ]
  • 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}{4}} f\left(\left((g(x))^2-h(x)\right)\right)
=f\left(\lim _{x \rightarrow \frac{\pi}{4}}\left((g(x))^2-h(x)\right)\right)
=f\left(\lim _{x \rightarrow \frac{\pi}{4}}(g(x))^2-\lim _{x \rightarrow \frac{\pi}{4}} h(x)\right)
=f\left(\left(\lim _{x \rightarrow \frac{\pi}{4}} g(x)\right)^2-\lim _{x \rightarrow \frac{\pi}{4}} h(x)\right)
=f\left(\left(\lim _{x \rightarrow \frac{\pi}{4}}(\sin x+\cos x)\right)^2-\lim _{x \rightarrow \frac{\pi}{4}}(\sin x-\cos x)\right)
=f\left(\left(\sin \frac{\pi}{4}+\cos \frac{\pi}{4}\right)^2-\left(\sin \frac{\pi}{4}-\cos \frac{\pi}{4}\right)\right)
=f\left(\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)^2-\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)\right)
=f(2-0)
=f(2)
 

Posted by

Irshad Anwar

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