Question

Use graphs and tables to find the limit and identify any vertical asymptotes of the function. limit of 1 divided by the quantity x minus 1 squared as x approaches 1

Answer 1

Answer:

Limit: infinite

vertical asymptote: x=1

Step-by-step explanation:

To find the limit, you can graph 1 divided by x minus 1 squared on desmos and  set x=1, and you should see that the function never touches x=1 making it infinite.

To find the vertical asymptote set the denominator equal to 0 then solve, so (x-1)^2=0, take the square root of both sides, x-1=0, add 1, x=1.

Answer 2

$\bold{\lim_{x \to 1} \frac{1}{(x-1)^2}=+\infty}$ and the vertical asymptote = 1

What is vertical asymptotes?

" Vertical asymptotes occur at the values where a rational function has a denominator of 0."

For given question,

We have been given a limit of function.

We need to find the value of the limit and the vertical asymptote for given function.

limit of 1 divided by the quantity x minus 1 squared as x approaches 1

$\Rightarrow \lim_{x \to 1} \frac{1}{(x-1)^2}$

To find the vertical asymptotes, we must set the denominator equal to

0 and solve the following equation:

⇒ (x - 1)² = 0

⇒ x - 1 = 0

⇒ x = 1

Thus, x = 1 is the vertical asymptote.

Consider the given function f(x) = 1/(x - 1)²

The graph of given function and its vertical asymptote is as shown below.

From the following table, we can observe that as x approaches 1 from left then function value tends to + infinity and if x values approaches 1 from the right, then function value tends to +infinity.

The left hand limit of given function is,

$\Rightarrow \lim_{x \to 1^-} \frac{1}{(x-1)^2} = +\infty$                        ......................(1)

The right hand limit of given function is,

$\Rightarrow \lim_{x \to 1^+} \frac{1}{(x-1)^2} = +\infty$                        ......................(2)

From (1) and (2) the limit of given function would be,

$\Rightarrow \lim_{x \to 1} \frac{1}{(x-1)^2} = +\infty$

Therefore, $\lim_{x \to 1} \frac{1}{(x-1)^2}=+\infty$ and the vertical asymptote = 1

Learn more about vertical asymptotes here:

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