For the function f whose graph is given, state the following. (a) limx →∞ f(x) (b) limx →- ∞ f(x

step 1 So in this problem were asked a series of limits and they were asked for the equations of the as

For the function f whose graph is given, state the...

For the function $ f $ whose graph is given, state the following.
(a) $ \displaystyle \lim_{x \to \infty} f(x) $
(b) $ \displaystyle \lim_{x \to - \infty} f(x) $
(c) $ \displaystyle \lim_{x \to 1} f(x) $
(d) $ \displaystyle \lim_{x \to 3} f(x) $
(e) The equations of the asymptotes

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur at points where the function becomes unbounded or grows infinitely in magnitude, typically due to a division by zero or a discontinuity in the function. They indicate the x-values at which the function is undefined and the graph 'blows up' towards positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behaviour of a function as the input variable tends to positive or negative infinity. If the function approaches a fixed value as x becomes extremely large or extremely small, that constant is defined as the horizontal asymptote of the function.

One-sided Limits

One-sided limits consider the behaviour of a function as the input approaches a specified value from only one side—either from the left or from the right. This is critical for analyzing functions that have different behaviours on either side of a given point, which in turn affects continuity and the existence of limits.

Limits

Limits describe the value that a function approaches as its input approaches a particular point (which can be a number or infinity). This concept is fundamental in calculus as it provides the basis for understanding continuity, derivatives, and integrals.

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