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Find the limit of the function algebraically. limit as x approaches zero of quantity x squared plus two x divided by x to the fourth power.


Please give a full explanation!

Respuesta :

PunIntended

Answer:

DNE (does not exist)

Step-by-step explanation:

Our function is: [tex]f(x)=\frac{x^2+2x}{x^4}[/tex]. We want to find the limit of this as x approaches 0. The first thing we would want to do is to substitute 0 in for x. But when we do that, we get 0/0, which is undefined:

[tex]\frac{0^2+2*0}{0^4}=\frac{0}{0}[/tex]

Let's divide the numerator and denominator both by x:

[tex]\frac{x^2+2x}{x^4}=\frac{x+2}{x^3}[/tex]

Now substitute 0 in again:

[tex]\frac{0+2}{0^3}=\frac{2}{0}[/tex]

Because we have a number divided by 0, this cannot exist. If we graph this function (see attachment), we'll also see that the graph diverges at x = 0, so the limit does not exist.

Ver imagen PunIntended
amna04352

Answer:

Limit doesn't exist

Step-by-step explanation:

As x --> 0,

(x² + 2x)/x⁴

From the LHS:

Let x = -0.1, (x² + 2x)/x⁴ = -1900

From the RHS:

Let x = 0.1, (x² + 2x)/x⁴ = 2100

Therefore the limit doesn't exist