Question

Limit x tends to y tanx- tany/x-y

Answer 1

Have you read "Sandwich Theorem"

If yes then you must be Knowing, lim (x→0) sinx ≤ lim (x→0) x ≤ lim (x→0) tanx

So it means:

★ lim (x→0) sinx = lim (x→0) x = lim (x→0) tanx

You must also be Knowing from this theorem we can write:

lim (x→0) sinx = lim (x→0) x
So, lim (x→0) [(sin x)/x] = 1
Similarly, lim (x→0) [(tan x)/x] = 1

Use this theorem.

Also, tan(x-y) = (tan x - tan y)/(1 - tanx tany)
Cross multiplying:
(1 - tanx tany) tan(x-y) = (tan x - tan y)
Replace above in the question, we get

→ lim (x→y) [(1 - tanx tany) tan(x-y)/(x-y)]
→ lim (x-y→0) [(1 - tanx tany) tan(x-y)/(x-y)]
→ lim (x-y→0) [(1 - tanx tany) (1)]
→ lim (x→y) [(1 - tanx tany)]
Now put

→ (1 - tan² y)

{ lim (x→y) = lim (x-y→0)
both are equal, you can choose to write any. }

Thankyou!!!

Answer 2

Answer:

Hey mate Here is your Answer ☺️