Limit as x approaches 0 of (square root(4+x^4)-2)/x^4)

Thread starter famallama
Start date Sep 6, 2008
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In summary, the limit as x approaches 0 of (square root(4+x^4)-2)/x^4) represents the slope of the tangent line at the point (0,2) on the curve y = square root(4+x^4). To calculate this limit, the quotient rule for limits can be used. The value of the limit is equal to 1/4 and it cannot be evaluated using direct substitution. The graph of the function approaches a vertical tangent line at the point (0,2), meaning it gets steeper as x approaches 0.

Sep 6, 2008

famallama

limit as x approaches 0 of (square root(4+x^4)-2)/x^4)
it says to solve algebraically by rationalizing the numerator

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Sep 6, 2008

Dick

Science Advisor

Homework Helper

To rationalize it multiply numerator and denominator by sqrt(4+x^4)+2. Use (a+b)*(a-b)=a^2-b^2.

1. What does the limit as x approaches 0 of (square root(4+x^4)-2)/x^4) represent?

The limit as x approaches 0 of (square root(4+x^4)-2)/x^4) represents the slope of the tangent line at the point (0,2) on the curve y = square root(4+x^4).

2. How do you calculate the limit as x approaches 0 of (square root(4+x^4)-2)/x^4)?

To calculate the limit as x approaches 0 of (square root(4+x^4)-2)/x^4), we can use the quotient rule for limits. This involves finding the limit of the numerator and the limit of the denominator separately, and then dividing the two limits.

3. What is the value of the limit as x approaches 0 of (square root(4+x^4)-2)/x^4)?

The value of the limit as x approaches 0 of (square root(4+x^4)-2)/x^4) is equal to 1/4.

4. Can the limit as x approaches 0 of (square root(4+x^4)-2)/x^4) be evaluated using direct substitution?

No, the limit as x approaches 0 of (square root(4+x^4)-2)/x^4) cannot be evaluated using direct substitution because it would result in an indeterminate form of 0/0, which is undefined.

5. How does the graph of y = (square root(4+x^4)-2)/x^4) behave near the point (0,2)?

The graph of y = (square root(4+x^4)-2)/x^4) approaches a vertical tangent line at the point (0,2). This means that as x gets closer and closer to 0, the graph gets steeper and steeper near the point (0,2).