Q55 / 150 · MHT CET 2024 (11 May Shift 1) · Mathematics

$\lim _{x \rightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}\right)=$

  • A
    $\frac{-16}{3}$
  • B
    $\frac{8}{3}$
  • Correct answer
    C
    $\frac{-8}{3}$
  • D
    $\frac{16}{3}$

Solution

Let $\begin{aligned} L & =\lim _{x \rightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}\right) \\ & =\lim _{x \rightarrow 2}\left(\frac{5^x+\frac{5^3}{5^x}-30}{\frac{5^3}{5^x}-\left(5^x\right)^{\frac{1}{2}}}\right)\end{aligned}$ $\begin{aligned} & \text { Let } t=5^x \\ \therefore \quad x & \rightarrow 2 \Rightarrow t \rightarrow 25 \\ \therefore \quad L & =\lim _{t \rightarrow 25}\left(\frac{t+\frac{125}{t}-30}{\frac{125}{t}-\sqrt{t}}\right) \\ & =\lim _{t \rightarrow 25}\left(\frac{t^2-30 t+125}{25 \sqrt{25}-t \sqrt{t}}\right) \\ & =\lim _{t \rightarrow 25}\left(\frac{(t-25)(t-5)}{25^{\frac{3}{2}}-t^{\frac{3}{2}}}\right)\end{aligned}$ $\begin{aligned} & =\lim _{\mathrm{t} \rightarrow 25} \frac{\mathrm{t}-5}{-\left(\frac{25^{\frac{3}{2}}-\mathrm{t}^{\frac{3}{2}}}{25-\mathrm{t}}\right)} \\ & =\frac{25-5}{-\frac{3}{2}(25)^{\frac{1}{2}}} \\ & =\frac{-8}{3}\end{aligned}$

Hence, the correct answer is option: 'C'.

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