Question

lim x tends to pi by 4 2-cosec^2 x/1 - cotx​

Answer 1

Answer:

lim x → π/ 4 [ 2 - cosec²x/ 1 - cotx] = 2

Step-by-step explanation:

This question is actually very easy if you are able to recall cosec²x = 1 + cotx. Now put this value in the question,

L = lim x → π/4 [2 - (1 + cot²x)/ 1 - cot]

L = lim x → π/4 [ 2 - 1 - cot²x/ 1 - cotx]

L = lim x → π/4 [ 1 - cot²x/ 1 - cotx]

We can write 1 as 1² because it makes no difference,

L = lim x → π/4 [1² - cot²x/ 1 - cotx]

Recall your childhood times now, you must have seen this somewhere a² - b² = (a + b) (a + b), using this,

L = lim x → π/4 [ (1+ cotx) (1 - cotx) / 1 - cotx]

L = lim x → π/4 [ 1 + cotx]

Now substitute the limiting value,

L = 1 + cot (π/4)

L = 1 + 1

L = 2.

Answer 2

Question:-

$\sf \large \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \: \sf \large \frac{2 - cosec {}^{2} x}{1 - cot \: x} .$

Given:-

The limit

$\sf \large \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \: \sf \large \frac{2 - cosec {}^{2} x}{1 - cot \: x} .$

To Find:-

• Need to evaluate the given function.

Solution:-

$\sf \large when \: x = \frac{\pi}{4} ,the \: expansion \\ \sf \large\sf \large \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \: \sf \large \frac{2 - cosec {}^{2} x}{1 - cot \: x} \: assumes \: the \: form \bigg( \frac{0}{0} \bigg ).$

$\sf \large \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \: \sf \large \frac{2 - cosec {}^{2} x}{1 - cot \: x} = \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \bigg[ \sf \large \frac{2 -(1 + cot {}^{2} x)}{1 - cot \: x)} \bigg ] = \sf \large\sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \sf \large \bigg[ \frac{1 - cot {}^{2}x }{1 - cot \: x} \bigg ]$

Upon expansion, we get

$\sf \large = \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \sf \large \bigg[ \frac{(1 - cot \: x)(1 + cot \: x)}{(1 - cot \: x)} \bigg ]$

Now substituting the value of x, we get

$\sf \large = 1 + cot \bigg( \frac{\pi}{4} \bigg) \\ \\ \sf \large = 1 + 1 \\ \\ \sf \large = 2.$

Answer:-

$\sf \large \color{red}\sf \large \sf\tt{\displaystyle \sf \lim_{x \to \frac{\pi}{4} }} \: \sf \large \frac{2 - cosec {}^{2} x}{1 - cot \: x} = 2.$

Hope you have satisfied. ⚘