Evaluate the following limit : limx→0[15x-5x-3x+1x⋅sinx] - Mathematics and Statistics

प्रश्न

Evaluate the following limit :

$lim_{x \to 0} [\frac{15^{x} - 5^{x} - 3^{x} + 1}{x \cdot \sin x}]$

बेरीज

उत्तर

$lim_{x \to 0} [\frac{15^{x} - 5^{x} - 3^{x} + 1}{x \sin x}]$

= $lim_{x \to 0} \frac{5^{x} \cdot 3^{x} - 5^{x} - 3^{x} + 1}{x \sin x}$

= $lim_{x \to 0} \frac{5^{x} (3^{x} - 1) - (3^{x} - 1)}{x \sin x}$

= $lim_{x \to 0} \frac{(3^{x} - 1) (5^{x} - 1)}{x \sin x}$

= $lim_{x \to 0} \frac{(\frac{3^{x} - 1}{x}) (\frac{5^{x} - 1}{x})}{(\frac{\sin x}{x})}$ ...[∵ x → 0, ∴ x ≠ 0]

= $\frac{(\log 3) (\log 5)}{1} ... [∵ lim_{x \to 0} \frac{a^{x} - 1}{x} = \log a]$

= (log 3) (log 5).

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Limits of Exponential and Logarithmic Functions

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q III. (3)Q III. (2)Q III. (4)

पाठ 7: Limits - Exercise 7.6 [पृष्ठ १५४]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board

पाठ 7 Limits
Exercise 7.6 | Q III. (3) | पृष्ठ १५४

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