Compute the limit:
lim (tan x – tan a)/tan(x – a)
x → a
without l'hospital's rule.
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Step-by-step explanation:

we are given a limit

and said to compute without L'Hopitâl rule

if we substitute a for x directly we'd get

which is indeterminate

so we have to do it differently

recall trigonometric indentity

using the identity we get

simplify complex fraction:

now we can substitute a for x:

simplify multiplication:

recall trigonometric indentity:

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional