Question

Please find the general limit of the following function:

Answer 1

Answer:

The general limit exists at x = 9 and is equal to 300.

Step-by-step explanation:

We want to find the general limit of the function:

$\displaystyle \lim_{x \to 9}(x^2+2^7+(9.1\times 10))$

By definition, a general limit exists at a point if the two one-sided limits exist and are equivalent to each other.

So, let's find each one-sided limit: the left-hand side and the right-hand side.

The left-hand limit is given by:

$\displaystyle \lim_{x \to 9^-}(x^2+2^7+(9.1 \times 10))$

Since the given function is a polynomial, we can use direct substitution. This yields:

$=(9)^2+2^7+(9.1\times 10)$

Evaluate:

$300$

Therefore:

$\displaystyle \lim_{x \to 9^-}(x^2+2^7+(9.1 \times 10))=300$

The right-hand limit is given by:

$\displaystyle \lim_{x \to 9^+}(x^2+2^7+(9.1\times 10))$

Again, since the function is a polynomial, we can use direct substitution. This yields:

$=(9)^2+2^7+(9.1\times 10)$

Evaluate:

$=300$

Therefore:

$\displaystyle \lim_{x \to 9^+}(x^2+2^7+(9.1\times 10))=300$

Thus, we can see that:

$\displaystyle \lim_{x \to 9^-}(x^2+2^7+(9.1\times 10))=\displaystyle \lim_{x \to 9^+}(x^2+2^7+(9.1\times 10))=300$

Since the two-sided limits exist and are equivalent, the general limit of the function does exist at x = 9 and is equal to 300.

Answer 2

Step-by-step explanation:

Hey there!

Please look your required answer in picture.

Note: In left hand limit always take a smaller near number of the approaching number. For example as in the solution I took the 8.99,8.999 as it is smaller than 9 but very near to it.

And in right hand limit always take a smaller and just greater near number than the approaching number. For example, I took 9.01,9.001 which a just greater but very near to 9.

Hope it helps!