Limits, Continuity & Differentiability Question - None

Question

Let $f(x) = \dfrac{\tan x}{x}$ , then the value of $\lim_{x \to 0}\left([f(x)] + x^2\right)^{\frac{1}{\{f(x)\}}}$ is equal to
(where [.] and {.} denote greatest integer function and fractional part functions respectively)

Accepted Answer

Key Idea: Analyse $\lim_{x \to 0}\left([f(x)] + x^2\right)^{\frac{1}{\{f(x)\}}}$ near 0, then apply GIF and fractional part carefully.

Step 1: As $$x o 0^+$$ , since $$ an x > x$$ for small $$x > 0$$ :
$f(x) = \frac{ an x}{x} o 1^+$ So $$[f(x)] = 1$$ and $\{f(x)\} = f(x) - 1 o 0^+$

Step 2: The expression becomes:
$\lim_{x o 0}(1 + x^2)^{\frac{1}{\{f(x)\}}}$ This is $1^\infty$ form.

Step 3: Find $\{f(x)\}$ precisely using Taylor expansion:
$\frac{ an x}{x} - 1 = \frac{ an x - x}{x} \approx \frac{x^3/3}{x} = \frac{x^2}{3}$ So $\{f(x)\} \approx \dfrac{x^2}{3}$ , giving exponent $= \dfrac{3}{x^2}$ .

Step 4:
$\lim_{x o 0}(1+x^2)^{3/x^2} = e^3$

Answer: (B) $$e^3$$

Trap: Students compute

$[f(x)] = 1$

correctly but treat

(1 + x^2)^{1/\{f(x)\}}

as simply

e^{x^2 / \{f(x)\}}

without finding the exact rate of

\{f(x)\} o 0

. The Taylor expansion of

an x - x \approx x^3/3

is the key step most students miss.

Key Concept: GIF and fractional part in limits combined with

1^\infty

form

Question:

Step-by-Step Solution

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