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MHTCET 25 Apr 2025 Shift 1 Paper

Question : 133 of 150

Marks: +1, -0

\lim _{x \rightarrow 5} \frac{\sqrt{2-2 \cos \left(x^{2}-12 x+35\right)}}{(x-5)}=\ldots \ldots

Solution:

Concept:

Use

1-\cos\theta = 2\sin^2(\theta/2)

and

\sqrt{a^2} = |a|

Explanation:

Let

\theta = x^2 - 12x + 35 = (x-5)(x-7)

Then

2 - 2\cos\theta = 4\sin^2(\theta/2)

, so the numerator is

2|\sin(\theta/2)|

Put

u = \frac{\theta}{2} = \frac{(x-5)(x-7)}{2}

; the expression becomes

\frac{2|\sin u|}{x-5}

For

x \to 5^{-}

u > 0

and

|\sin u| = \sin u

, so

\frac{2\sin u}{x-5} = (x-7)\frac{\sin u}{u} \to (5-7)(1) = -2

For

x \to 5^{+}

u < 0

and

|\sin u| = -\sin u

, so

\frac{-2\sin u}{x-5} = -(x-7)\frac{\sin u}{u} \to -(5-7) = 2

Since the left-hand limit

-2

and the right-hand limit

2

are not equal, the two-sided limit does not exist.

Answer:

The limit does not exist; none of the given options is correct. Option B (

-2

) gives only the left-hand limit.

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