A thermometer is moved from a room where the temperature is $84$F to a freezer where the temperature is $9$F$.$ After $30$ seconds the thermometer reads $52$F$.$ Let T$($t$)$T$($t$)$ denote the temperature, in Fahrenheit, at time tt$,$ in seconds.

$($a$)$ Write a differential equation for T$($t$)$T$($t$).$ The only unknown should be T$($t$)$T$($t$).$

$($b$)$ What does the thermometer read after $2$ minutes?

$($c$)$ Compute the limit$.$