Ready for the 10 limits questions with answers?

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Remember to determine if the limit exists for the functions on the worksheet. If the limit does exist, write the value, if it does not, state why (jump, asymptote, etc.).

Background Information

Calculus has been around since the seventeenth century out of a need to solve two
important key geometric issues: finding tangent lines to curves (known as differential calculus) and calculating areas under curves (known as integral calculus).

The concept of the limit was not fully explained until the nineteenth century when French mathematician Augustin-Louis Cauchy provided us with this definition: “When the values successively attributed to the same variable approach a fixed value indefinitely, in such a way as to end up differing from it by as little as one could wish, this last value is called the limit of all the others. So, for example, an irrational number is the limit of the various fractions which provide values that approximate it more and more closely.” This was a verbal definition and was translated by J. Grabiner. When considering the notion of a limit. Think of a function ƒ and some fixed number n in the domain of ƒ Then consider a sequence of values in the domain of ƒ where the sequence of numbers gets nearer and nearer to n. What will then happen to the values of ƒ (x)? As these values also get nearer and nearer to the fixed number, then the number is called the limit of ƒ as x approaches a.

A function has a limit for a specific x-value if the function hones in on a specific height as x gets closer to the specific value from the left and the right. You will also need to understand asymptotes (vertical or horizontal), those imaginary lines that a function gets closer to as it goes up, down, right, or left towards infinity. It is easier to understand limits by looking at the example on the PDF as opposed to sort through explanations. Limits are used for both differential and integral calculus. Like so many things in math, there are definitely some limits that are worth memorizing.