Lim x- 0 amx-bn x sin k x

Limx #8594;0 #160;amx bnxsin #160;kx=limx #8594;0 #160;amx 1 bnx 1sin #160;kxDividing #160;the #160;numerator #160;and #160;the #160;denom ...

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Standard XII

Mathematics

Properties of Determinants

lim x→ 0 amx-...

Question

$\lim_{x \to 0} \frac{a^{m x} - b^{n x}}{\sin k x}$

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Solution

$\lim_{x \to 0} [\frac{a^{m x} - b^{n x}}{\sin (k x)}] = \lim_{x \to 0} [\frac{(a^{m x} - 1) - (b^{n x} - 1)}{\sin k x}] Dividing the numerator and the denomiantor by x: = \lim_{x \to 0} [\frac{m (\frac{a^{m x} - 1}{m x}) - n (\frac{b^{n x} - 1}{n x})}{k \times \frac{\sin k x}{k x}}] = \frac{(m \log a - n \log b)}{k \times 1} = \frac{1}{k} [\log {(a)}^{m} - \log {(b)}^{n}] = \frac{1}{k} \log (\frac{a^{m}}{b^{n}})$

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${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{s i n k x}$

${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{x}$

$Let f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a (x) & b (x) & c (x) m (x) & n (x) & l (x) g (x) & h (x) & k (x) \end{matrix} ∣ ∣ ∣ ∣ ∣ then$

$\int_{0}^{a} f (x) is ∣ ∣ ∣ ∣ ∣ \begin{matrix} \int_{0}^{a} a (x) & \int_{0}^{a} b (x) & \int_{0}^{a} c (x) \int_{0}^{a} m (x) & \int_{0}^{a} n (x) & \int_{0}^{a} l (x) \int_{0}^{a} g (x) & \int_{0}^{a} h (x) & \int_{0}^{a} k (x) \end{matrix} ∣ ∣ ∣ ∣ ∣$

Q. STATEMENT-1 :

{lim}_{x \to \infty} (\frac{1^{2}}{x^{3}} + \frac{2^{2}}{x^{3}} + \frac{3^{2}}{x^{3}} + . . . . . . . + \frac{x^{2}}{x^{3}}) = lim x \to \infty \frac{1^{2}}{x^{3}} + lim x \to \infty \frac{2^{2}}{x^{3}} + . . . . . + lim x \to \infty \frac{x^{2}}{x^{3}} = 0

STATEMENT-2 :

{lim}_{x \to a} (f_{1} (x) + f_{2} (x) + . . . . . + f_{n} (x)) = {lim}_{x \to a} f_{1} (x) + {lim}_{x \to a} f_{2} (x) + . . . . + {lim}_{x \to a} f_{n} (x)

, where

n \in N .

If both $lim x \to a f (x)$ and $lim x \to a g (x)$ and exist finitely and $lim x \to a g (x) = 0$ , then
$lim x \to a \frac{f (x)}{g (x)} = \frac{lim x \to a f (x)}{lim x \to a g (x)}$

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limx→0 amx-bnxsin kx

limx→0 amx-bnxsin kx=limx→0 amx-1-bnx-1sin kxDividing the numerator and the denomiantor by x:=limx→0 mamx-1mx-nbnx-1nxk×sin kxkx=m log a-n log bk×1=1k log am-log bn=1k log ambn

$\lim_{x \to 0} \frac{a^{m x} - b^{n x}}{\sin k x}$