Use the rules for limits to evaluate the following. (a) limx →0 (x^3-1)/(x^2+1) (b) limx →1 (x-1

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Use the rules for limits to evaluate the following. (a) limx...

Use the rules for limits to evaluate the following. (a) $\lim _{x \rightarrow 0} \frac{x^{3}-1}{x^{2}+1}$ (b) $\lim _{x \rightarrow 1} \frac{x-1}{x^{3}}-\frac{1}{-1}$ (c) $\lim _{x \rightarrow 1} \cos \left(\frac{\pi x}{x+1}\right)$ (d) $\lim _{x \rightarrow 0} x^{2} \cos \left(\frac{1}{x}\right)$

Use the rules for limits to evaluate the following.
(a) $\lim _{x \rightarrow 0} \frac{x^{3}-1}{x^{2}+1}$
(b) $\lim _{x \rightarrow 1} \frac{x-1}{x^{3}}-\frac{1}{-1}$
(c) $\lim _{x \rightarrow 1} \cos \left(\frac{\pi x}{x+1}\right)$
(d) $\lim _{x \rightarrow 0} x^{2} \cos \left(\frac{1}{x}\right)$

Key Concepts

Composition of Functions

When a function is composed of two or more functions, the limit of the composite function can be found by taking the limit of the inner function and then applying the outer function to that limit, assuming the outer function is continuous at the limit of the inner function. This is particularly useful when dealing with trigonometric functions like cosine, whose limits can be computed directly through their continuous nature.

Squeeze Theorem

The squeeze theorem is used for evaluating limits of functions that are difficult to handle directly, especially when the function involves an oscillatory component that is bounded. If a function is 'squeezed' between two other functions that share the same limit at a certain point, then the original function also has that limit. This technique is often applied when one factor tends to zero while the other is bounded, ensuring the product tends to zero.

Continuity

Continuity is the property of a function where the limit of the function as the variable approaches a certain point equals the function's value at that point. Many standard functions, such as polynomials, trigonometric functions, and rational functions (with nonzero denominator), are continuous on their domains. Continuity simplifies limit evaluation, as one can directly substitute the variable's value into the function.

Limit Laws

These are rules that allow the evaluation of limits by breaking up complex expressions into simpler parts. They include properties such as the limit of a sum being the sum of the limits, the limit of a product being the product of the limits, and the limit of a quotient being the quotient of the limits (provided the limit of the denominator is nonzero). This makes evaluating limits of rational functions or composite functions straightforward when the individual limits exist.

Direct Substitution

This is a technique for evaluating limits by substituting the value that the variable is approaching directly into the function. It works when the function is continuous at that point, meaning the function's value at that point is defined and equals the limit. This approach is typically used for polynomials and rational functions where the denominator does not vanish at the point of interest.

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