We see that the value when approaching from the right matches the value when approaching from the left, and also we see that the value converges to (blank). therefore, the limit exists and is given by lim tan(4x)/tan(5x) = (blank)

The question is the limit of [tan(4x) / tan(5x) ] as x->0.

Both blanks must have the same value.

Limit [tan(4x) / tan(5x) ] as x->0 = Limit of { [sin (4x) / cos (4x) ] / [sin (5x) / cos (5x) ] } as x->0 =

= [4x / 1] / [5x / 1] = 4x / 5x = 4/5

4/5