Evaluate the expression

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Determine if the integral converges or diverges

Improper Integrals

By inspection we can see that this integral will have serious problems. Even before taking limits we notice that the limits of integration coincide with undefined values of the integrand.

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Limit of a radical expression

We multiply by the conjugate of an expression quite often - typically when we expect that it will help us remove an unwanted radical from the denominator, or in this case an unwanted expression from the denominator. This example skips a step or two here and there but I hope that it's still clear.

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Limit Definition of Derivatives

Use the definition of derivatives to find the derivative of sqrt(x)

Solution:
Multiplying by a conjugate can be a useful strategy and it helps in this problem. We also notice that sqrt(2) is only defined for real numbers greater than or equal to 0 (for calculus). This means the absolute value of an expression is also greater than or equal to 0. In the denominator we must be careful not to multiply because factoring out the delta-x is easier.

Limit of an exponential function

Limits of exponential functions sometimes call for L'Hopital's Rule. In this example the limit is much easier to compute after we apply the rule.

Limit of a sequence

Limit of a sequence. We us the handy strategy of re-writing the limit in a more convenient way that does not change it's value. And by using the conjugate of the expression, we are able so simplify. Alternately we can apply the direct comparison test. 1/n is a sequence that we know converges, so we compare our sequence to it. After some algebra, we can see that the inequality will hold.