First slide

Introduction to limits

Question

$If m, n \in N, \lim_{x \to 0} \frac{{sinx}^{n}}{{(sinx)}^{m}} is$

Moderate

A

$1 . if n = m$
B

$0 . if n > m$
C

$\infty . if n < m$
D

$\frac{n}{m} . if n < m$

Solution

$L = \lim_{x \to 0} \frac{{sinx}^{n}}{{(sinx)}^{m}} = \lim_{x \to 0} \frac{\frac{{sinx}^{n}}{x^{n}} x^{n}}{\frac{{(sinx)}^{m}}{x^{m}} x^{m}} = \lim_{x \to 0} x^{n - m} If n = m, then L = {(a very small value near to zero)}^{exactly zero} = 1 If n > m, then \begin{array}{l} L = {(a very small value near to zero)}^{positive integer} = 0 \\ If n < m, then \\ L = {(a very small value near to zero)}^{negative integer} = \infty \end{array}$

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