How do you find the asymptotic distribution of the OLS estimator in a simple linear regression model with an intercept?

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How do I find the asymptotic distribution of $\beta_1$ in the model $y_i=\beta_0+\beta_1x_i+\epsilon_i$?

I am able to follow Cameron's derivation of a very similar model without an intercept, but when I try to use the same approach in my model, I run into issues because I can't get the intercept term to go away.

Any help would be much appreciated!