How to solve this limit question?

Question

lim = (14x + 7) / (2x+1)

x->-1/2

Multiply the denominator and numerator by the highest power in the denominator

lim = (14x + 7) . (1/x) / (2x+1) . (1/x)

x->-1/2

lim = ( (14x/x) + (7/x) ) / ( (2x/x) + (1/x) )

x->-1/2

lim = ( (14x/x) + (7/x) ) / ( (2x/x) + (1/x) )

x->-1/2

lim = ( (14x/x) + (7/x) ) / ( (2x/x) + (1/x) )

x->-1/2

lim = ( 14 + 7/x ) / ( 2 + 1/x )

x->-1/2

Therefore, lim = ( 14 + 7/(-0.5) ) / ( 2 + 1/(-0.5) )

x->-1/2

= 10.5 / 0

meaning limit doesnt exist but my teacher got 7.

How do you get the right answer?

Answer 1

(14x + 7) / (2x+1)

=7(2x+1)/(2x+1) = 7.

Why do all that stuff when easy methods are there.