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  • Evaluate the limit <img alt="\lim_{x\rightarrow -1}f\left ( g\left ( x \right )+h\left ( x \right ) \right )" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%20-1%7Df%

Evaluate the limit \lim_{x\rightarrow -1}f\left ( g\left ( x \right )+h\left ( x \right ) \right ) where h\left ( x \right )= \left (\frac{x^{2}}{4}-14x+30 \right ),g\left ( x \right )= \left ( x^{2}-7x+12 \right ).

Option: 1

f\left ( 64.25 \right )


Option: 2

f\left ( -64.25 \right )


Option: 3

f\left ( 64 \right )


Option: 4

f\left ( 65 \right )


Answers (1)

best_answer

The limit stated here is \lim_{x\rightarrow -1}f\left ( g\left ( x \right )+h\left ( x \right ) \right )\quad \dots\left ( i \right )

where the following data is provided

h\left ( x \right )= \left (\frac{x^{2}}{4}-14x+30 \right ) \quad \dots\left ( ii \right )
g\left ( x \right )= \left ( x^{2}-7x+12 \right ) \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 “Sum 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 ].


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-1} f(g(x)+h(x))
=f\left(\lim _{x \rightarrow-1}(g(x)+h(x))\right)
=f\left(\lim _{x \rightarrow-1} g(x)+\lim _{x \rightarrow-1} h(x)\right)
=f\left(\lim _{x \rightarrow-1}\left(x^2-7 x+12\right)+\lim _{x \rightarrow-1}\left(\frac{x^2}{4}-14 x+30\right)\right)
=f\left(\left((-1)^2-7 \times(-1)+12\right)+\left(\frac{(-1)^2}{4}-14 \times(-1)+30\right)\right)
=f(20+44.25)
=f(64.25)


 

Posted by

Divya Prakash Singh

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