Assume that limθ→−1 f(θ)exists and θ2+θ−2θ+3≤f(θ)θ2≤θ2+2θ−1θ+3holds for certain interval containing the point θ=-1, then limθ→−1 f(θ)θ2 , is

We have,

$lim_{θ \to - 1} \frac{θ^{2} + θ - 2}{θ + 3} = - 1$ and $lim_{θ \to - 1} \frac{θ^{2} + 2 θ - 1}{θ + 3} = - 1$

$∴ \frac{θ^{2} + θ - 2}{θ + 3} \leq \frac{f (θ)}{θ^{2}} \leq \frac{θ^{2} + 2 θ - 1}{θ + 3} \Rightarrow lim_{θ \to - 1} \frac{f (θ)}{θ^{2}} = - 1$

Assume that $lim_{θ \to - 1} f (θ)$ exists and $\frac{θ^{2} + θ - 2}{θ + 3} \leq \frac{f (θ)}{θ^{2}} \leq \frac{θ^{2} + 2 θ - 1}{θ + 3}$ holds for certain interval containing the point $θ = - 1$ , then $lim_{θ \to - 1} \frac{f (θ)}{θ^{2}}$ , is