Consistency and Unbiasedness

Question

Suppose we want to estimate the parameters of the model

$$y_i = β⁠_1 + \frac{1}{β⁠_2}x_i + ​ɛ​_i$$

under the standard assumptions for OLS. We assume $β_2 > 0$. We run OLS on the regression

$$y_i = δ_1 + δ_2x_i + ​ɛ​_i$$

to obtain $\hat\delta_2$ and then use the estimator $\hat\beta_2 = \frac{1}{\hat\delta_2}$

1. Is $\hat\beta_2$ unbiased? If not, what is the sign of the bias?
2. Is $\hat\beta_2$ consistent?

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My Approach

First, to show that $\hat\beta_2$ is unbiased, i derived

$$\hat\delta_2 = \frac{cov(y_i,x_i)}{var(x_i)}$$ $$=\frac{cov(β⁠_1 + \frac{1}{β⁠_2}x_i + ​ɛ​_i,x_i)}{var(x_i)}$$ $$=\frac{cov(β⁠_1,x_i) + cov(\frac{1}{β⁠_2}x_i,x_i) + cov(​ɛ​_i,x_i)}{var(x_i)}$$ $$=\frac{cov(β⁠_1,x_i) +\frac{1}{β⁠_2}var(x_i) + cov(​ɛ​_i,x_i)}{var(x_i)}$$

Under standard OLS assumptions we have $cov(β⁠_1,x_i)=0$ and $cov(​ɛ​_i,x_i)=0$. We then get

$$\hat\delta_2 = \frac{1}{β⁠_2} => \hat\beta_2 = \frac{1}{\hat\delta_2} = \beta_2$$

$=> \hat\beta_2$ is unbiased as $E[\hat\beta_2]-\beta_2 = 0$

Second, as unbiasedness does not imply consistency, i am not sure how to proceed whether $\beta_2$ is consistent. I know that consistency further need LLN and CLT, but i am not sure how wo apply these two theorems.