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The function f(x) is defined as f(x) = (tan x + tan x) / sin x for x ≠ 0, and f(0) = a. Find the value of 'a' that makes the function continuous at x=0.

Mathematics

ScanSolve AI Answer

Answer

2\text{2}

Step-by-step solution

4 steps
  1. Understand the condition for continuity.

    For a function f(x)f(x) to be continuous at x=0x=0, the limit of the function as xx approaches 0 must be equal to the function's value at x=0x=0.
    Given f(0)=af(0) = a, we must have:

    a=lim⁡x→0f(x)a = \lim_{x \to 0} f(x)
  2. Simplify the function f(x)f(x).

    The given function is f(x)=ظا س+ظا سجا سf(x) = \frac{\text{ظا س} + \text{ظا س}}{\text{جا س}} for x≠0x \neq 0.
    This can be written as:

    f(x)=2tan⁡xsin⁡xf(x) = \frac{2 \tan x}{\sin x}
  3. Evaluate the limit.

    Substitute the simplified function into the limit expression:

    a=lim⁡x→02tan⁡xsin⁡xa = \lim_{x \to 0} \frac{2 \tan x}{\sin x}

    Recall that tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}. Substitute this identity into the expression:

    a=lim⁡x→02(sin⁡xcos⁡x)sin⁡xa = \lim_{x \to 0} \frac{2 \left(\frac{\sin x}{\cos x}\right)}{\sin x}

    Since x≠0x \neq 0, sin⁡x≠0\sin x \neq 0, so we can cancel sin⁡x\sin x from the numerator and denominator:

    a=lim⁡x→02cos⁡xa = \lim_{x \to 0} \frac{2}{\cos x}
  4. Substitute x=0x=0 into the simplified limit expression.

    a=2cos⁡(0)a = \frac{2}{\cos(0)}

    We know that cos⁡(0)=1\cos(0) = 1.

    a=21a = \frac{2}{1} a=2a = 2

    The value of aa is 2.
    Comparing this with the given options, option 2 is 2.

    The final answer is 2\text{2}.

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