Answer:
[tex]\lim_{x \to \infty} \frac{x}{x}=1[/tex]
Calculating the limit by product rule is WRONG
Step-by-step explanation:
Indeterminate Forms:
Indeterminate forms is an expression involving 2 functions whose limit cannot be determined by limits of individual functions. If we calculate the limits by general rules used for calculating the limits, we will not have a clear answer.
Examples of indeterminate forms are:
[tex]\frac{0}{0}, \frac{\infty}{\infty} , 0\cdot\infty, 1^{\infty},\infty-\infty, 0^0,\infty^0[/tex]
To find the limits of such forms, we have to use L'hospital rule, which states that if:
[tex]\lim_{x \to \infty} \frac{f(x)}{g(x)} = Indeterminate form \\ Then\\\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}[/tex]
Solve the question:
[tex]\lim_{x \to \infty} \frac{x}{x}= \lim_{x \to \infty} x\cdot \lim_{x \to \infty} \frac{1}{x} \\ \lim_{x \to \infty} \frac{x}{x}= =\infty\cdot\frac{1}{\infty} \\ \lim_{x \to \infty} \frac{x}{x}=\infty\cdot0[/tex]
As it is an INDETERMINATE FORM, we cannot calculate its limit by product rule. We have to use L'Hospital Rule:
[tex]\lim_{x \to \infty} \frac{x}{x}= \lim_{x \to \infty} \frac{d(x)/dx}{d(x)/dx}\\\lim_{x \to \infty} \frac{x}{x}=\lim_{x \to \infty} \frac{1}{1}\\\lim_{x \to \infty} \frac{x}{x}=1[/tex]
