Graph of the function y = f(x) is shown in the adjacent figure four limits ℓ1, ℓ2, ℓ3 and ℓ4 are given as: ℓ1 = \underset{x \rightarrow 0^{+}}{Lim} \textrm{ } \left(tan\right)^{- 1} \left[\right. f \left(\right. x \left.\right) - 1 \left]\right. ℓ2 : \underset{x \rightarrow 0^{-}}{Lim} \textrm{ } \left(sec\right)^{- 1} \left[\right. f \left(\right. x \left.\right) - f \left(\right. x^{2} \left.\right) \left]\right. ℓ3 : \underset{x \rightarrow 1^{-}}{Lim} \textrm{ } f \left(\right. x^{2} - x \left.\right) ℓ4 : \underset{x \rightarrow 0^{-}}{Lim} \left(cos\right)^{- 1} \left(\right. \textrm{ } f \left(\right. x^{3} \left.\right) - f \left(\right. x \left.\right) \textrm{ } \left.\right) Then find the value of \frac{\left(\text{l}\right)_{2} \left(\textrm{ }\text{l}\right)_{3}}{\left(\text{l}\right)_{1} + \left(\text{l}\right)_{4}} [Note: [k] denotes the greatest integer less than or equal to k.]

Engineering

Mathematics

Introduction to Limit

Question

Graph of the function y = f(x) is shown in the adjacent figure four limits ℓ₁, ℓ₂, ℓ₃ and ℓ₄ are given as:

ℓ₁ = $\underset{x \to 0^{+}}{Lim} \tan^{- 1} [f (x) - 1]$

ℓ₂ : $\underset{x \to 0^{-}}{Lim} \sec^{- 1} [f (x) - f (x^{2})]$

ℓ₃ : $\underset{x \to 1^{-}}{Lim} f (x^{2} - x)$

ℓ₄ : $\underset{x \to 0^{-}}{Lim} \cos^{- 1} (f (x^{3}) - f (x))$

Then find the value of $\frac{l_{2} l_{3}}{l_{1} + l_{4}}$

[Note: [k] denotes the greatest integer less than or equal to k.]

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$l_{1} = \frac{- π}{4}; l_{2} = \sec^{- 1} 2 = \frac{π}{3}; l_{3} = 3; l_{4} = \frac{π}{2}$

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