First slide

Continuity

Question

$If f (x) = \{\begin{cases} \frac{8^{x} - 4^{x} - 2^{x} + 1}{x^{2}}, x > 0 \\ e^{x} sinx + πx + λ \ln 4, x \leq 0 \end{cases}$ is continuous at x = 0, then the value of $λ$ is

Moderate

A

$4 \log_{e} 2$
B

$2 \log_{e} 2$
C

$\log_{e} 2$
D

none of these

Solution

$\begin{array}{l} f (0) = 0 + 0 + λ \ln 4 = λln 4 (1) \\ R . H . L . = \lim_{x \to 0 +} f (x) = \lim_{h \to 0} f (0 + h) \\ = \lim_{h \to 0} \frac{8^{h} - 4^{h} - 2^{h} + 1}{h^{2}} \\ = \lim_{h \to 0} \frac{(4^{h} - 1) (2^{h} - 1)}{h . h} \\ = \lim_{h \to 0} (\frac{4^{h} - 1}{h}) . . (\frac{2^{h} - 1}{h}) \\ = \ln 4 \ln 2 (2) \\ ∴ f (0) = R . H . L . \\ ∴ λ = \ln 2 \end{array}$

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