Function Continuity Condition
A function $f(x)$ is continuous at $x=a$ if $ \lim_{x \to a} f(x) = f(a) $. For the given function $f(x)$, we need $f(0) = \lim_{x \to 0} f(x)$.
Calculating the Limit of f(x) at x = 0
The function is $ f(x) = \frac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x} $. Direct substitution yields the indeterminate form $ \frac{0}{0} $. We evaluate the limit using standard calculus methods.
Method: Limit Decomposition
Split the function into two parts:
- $ \text{Part 1:} \lim_{x \to 0} \frac{e^x\left(e^{\tan x - x} - 1\right)}{\tan x - x} $
- $ \text{Part 2:} \lim_{x \to 0} \frac{\log_e(\sec x + \tan x) - x}{\tan x - x} $
Evaluating Part 1
Let $u = \tan x - x$. As $x \to 0$, $u \to 0$. The limit becomes $ \lim_{x \to 0} e^x \cdot \frac{e^u - 1}{u} $. Using the standard limit $ \lim_{u \to 0} \frac{e^u - 1}{u} = 1 $, Part 1 evaluates to $ e^0 \cdot 1 = 1 $.
Evaluating Part 2
Evaluating $ \lim_{x \to 0} \frac{\log_e(\sec x + \tan x) - x}{\tan x - x} $ using limit properties (such as L'Hôpital's Rule or Taylor series expansions) yields a value of 1.
Combining the Results
The total limit is the sum of the limits of the two parts:
$ \lim_{x \to 0} f(x) = 1 + 1 = 2 $
Thus, for the function to be continuous at $x=0$, $f(0)$ must be equal to this limit.
$ f(0) = 2 $
Similar Questions
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Important Questions from Differential Calculus
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_e x| - |x - 1|$:
(I) $f$ is differentiable at all $x > 0$.
(II) $f$ is increasing in $(0, 1)$.
(III) $f$ is decreasing in $(1, \infty)$.
Then. - If the domain of the function $f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $\alpha + \beta + \gamma + \delta$ is equal to
Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
$f(x) = \begin{cases} b^2 \sin \left( \frac{\pi}{2} \left[ \frac{\pi}{2} (\cos x + \sin x) \cos x \right] \right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$
is continuous at $x = 0$, then $a^2 + b^2$ is equal to
- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
