Evaluate the following limit: _ x 4 1- x- 4 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$
If $\text{x}\rightarrow\frac{\pi}{4},$ then $\text{x}-\frac\pi4\rightarrow0$
Let $\text{x}-\frac\pi4=\text{y}\Rightarrow\text{y}\rightarrow0$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\tan\big(\text{y}+\frac{\pi}{4}\big)}{\text{y}}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\Bigg(\frac{\tan\text{y}+\tan\frac{\pi}{4}}{1-\tan\text{y}\tan\frac{\pi}{4}}\Bigg)}{\text{y}}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{(1-\tan\text{y}-\tan\text{y}-1)}{\text{y}(1-\tan\text{y})}$ $\Big[\because\tan\frac{\pi}{4}=1\Big]$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{(-2\tan\text{y})}{(1-\tan\text{y})}$
$=-2\lim\limits_{\text{y}\rightarrow{0}}\frac{\tan\text{y}}{\text{y}}\times\frac{1}{\lim\limits_{\text{y}\rightarrow{0}}(1-\tan\text{y})}$
$=-2\times1\frac{1}{(1-0)}$ $\Big[\because\lim\limits_{\theta\rightarrow{0}}\frac{\sin\theta}{\theta}=1\Big]$
$=-2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Limits→Limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only?

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

In drilling world’s deepest hole it was found that the temperature T in degree celcius, x km below the earth’s surface was given by $\text{T}=30^\circ+25^\circ(\text{x}-3), 3\leq\text{x}\leq15$At what depth will the temperature be between 155°C and 205°C?

Prove that: $\tan36^\circ+\tan9^\circ+\tan36^\circ+\tan9^\circ=1$

Give an example of a statement P(n) which is true for all $\text{n}\geq4$ but P(1), P(2) and P(3) are not true. Justify your answer.

If $\frac{\text{z}-1}{\text{z}+1}$ is purely imaginary number $(\text{z}\neq-1),$ find the value of $|\text{z}|.$

The Sum of first three terms of a G.P. is $16$ and the Sum of the next three term is $128.$ determine the first term, the common ratio and the Sum to n terms of the G.P.

If P(n) is the statement $"2^n \geq 3n"$ and if $P(r)$ is true, prove that $P(r + 1)$ is true.

Find the mean variance and standard deviation for the following data:
$15, 22, 27, 11, 9, 21, 14, 9.$

Out of $25$ members in a family, $12$ like to take tea, $15$ like to take coffee and $7$ like to take coffee and tea both. How many like
$i.$ at least one of the two drinks
$ii.$ only tea but not coffee
$iii.$ only coffee but not tea
$iv.$ neither tea nor coffee