If the function f(x)=tanx−sinxtan(tanx)−sin(sinx) is continuous at x=0, then f(0) is equal to:
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Step-by-Step Solution
Step 1: Apply continuity condition
For a function to be continuous at a point, the function's value at that point must be equal to its limit as x approaches that point. Since f(x) is continuous at x=0, we need to find the limit of f(x) as x→0 to determine f(0).
Step 2: Substitute y=tanx and z=sinx
To simplify the expression, let's introduce new variables. Let y=tanx and z=sinx. As x→0, both y→0 and z→0. This substitution helps in recognizing a standard limit form.
Step 3: Rewrite the expression using Taylor series expansion
We use the Taylor series expansions for tany and sinz around 0. This allows us to approximate the functions with polynomials, which simplifies the limit calculation. We only need terms up to y3 and z3 for this problem.
Step 4: Substitute expansions and simplify
Substitute the Taylor series expansions back into the expression. We then separate the term (y-z)/(y-z) which simplifies to 1, and handle the remaining cubic terms.
Step 5: Evaluate the limit using y=x+3x3+O(x5) and z=x−6x3+O(x5)
Now, substitute back y=tanx and z=sinx. We use their Taylor expansions around x=0: tanx=x+3x3+O(x5) and sinx=x−6x3+O(x5). Notice that y−z=(x+3x3)−(x−6x3)=3x3+6x3=63x3=2x3. Also, for small x, y3≈x3 and z3≈x3. Substituting these into the expression, we find the limit to be 2.