# How do you find the limit #((1-x)^(1/4)-1)/x# as #x->0#?

According to the binomial expansion we have

Then

Another approach is rationalizing

By signing up, you agree to our Terms of Service and Privacy Policy

Multiply by the conjugate of the numerator over itself.

So

By signing up, you agree to our Terms of Service and Privacy Policy

To find the limit of ((1-x)^(1/4)-1)/x as x approaches 0, we can use algebraic manipulation and the limit definition.

First, let's simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is ((1-x)^(1/4)+1):

((1-x)^(1/4)-1)/x * ((1-x)^(1/4)+1)/((1-x)^(1/4)+1)

This simplifies to:

((1-x)^(1/4))^2 - 1^2 / x * ((1-x)^(1/4)+1)

Simplifying further:

(1-x)^(1/2) - 1 / x * ((1-x)^(1/4)+1)

Now, we can apply the limit definition. As x approaches 0, we substitute 0 into the expression:

(1-0)^(1/2) - 1 / 0 * ((1-0)^(1/4)+1)

Simplifying:

1^(1/2) - 1 / 0 * (1^(1/4)+1)

This further simplifies to:

1 - 1 / 0 * (1+1)

Since division by 0 is undefined, the limit does not exist.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you find the points of continuity of a function?
- How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x->oo#?
- How do you find #\lim _ { x \rightarrow 3} \frac { x ^ { 3} - 27} { x + 5}#?
- How do you find the limit of #(cos x)^(1/x^2)# as x approaches 0?
- How do you find the limit of #x Tan(9/x)# as x approaches infinity using l'hospital's rule?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7