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Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 0}\left( \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \right)$

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$-1 / 2$

03:31

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

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04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Evaluate the limit, if it …

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Find the limit (if it exis…

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to evaluate this limit so that we can rewrite it into the limit. S. T. A purchase zero of one over T. Time squared of one plus t minus The square root of one plus T over tee times the square root of one plus T. Combining the two fractions. We would get limit. SD approaches zero of you have one minus squared of one plus teeth over T times the square root of one plus T. Now in here we would rationalize the numerator and we multiply the whole expression by one plus the square root of one plus T over one plus the square root of one busty which is the conjugate of the numerator. From here we would get limit esti approaches zero of You have one The square of the square of one plus T. Over. We have tee times the square root of one plus T times one plus the square of one plus T. Simplifying this, we would have limit ste approaches zero of one minus one plus T over Teeth and the squares of one plus T times one plus the square root of one plus T simply find further. We would get limit as T approaches zero of negative T over tee times square at the one plus T times one plus the square root of one plus T. And in here we can cancel out the tea and we would get limit S. T. A purchase zero of negative one over The square root of one plus T Times one plus the square root of one plastic evaluating a T. Called zero. We would get negative one over Squared of one plus 0 times one plus the square root of one plus zero, which is just -1/2. And so this is the value of the limits.

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