# How do you find #\lim _ { x \rightarrow 7} \frac { x ^ { 3} - 343} { x - 7}#?

Direct substitution does not work because it gives us 0/0, an indeterminate form.

We could try factorising this rational function so that we can possibly cancel out common terms.

This is the graph below. graph{(x^3-343)/(x-7) [-24.6, 25.38, 129.5, 154.49]}

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To find the limit (\lim_{x \to 7} \frac{x^3 - 343}{x - 7}), we can use direct substitution:

[\lim_{x \to 7} \frac{x^3 - 343}{x - 7} = \frac{7^3 - 343}{7 - 7}]

Simplify the expression:

[= \frac{343 - 343}{0}]

[= \frac{0}{0}]

This is an indeterminate form, which means we can't directly evaluate the limit by substituting. To resolve this, we can factorize the numerator as the difference of cubes:

[x^3 - 343 = (x - 7)(x^2 + 7x + 49)]

Now, the expression becomes:

[\lim_{x \to 7} \frac{(x - 7)(x^2 + 7x + 49)}{x - 7}]

Cancel out the common factor of (x - 7):

[= \lim_{x \to 7} (x^2 + 7x + 49)]

Now, we can directly substitute (x = 7) into the expression:

[= 7^2 + 7(7) + 49]

[= 49 + 49 + 49]

[= 147]

So, (\lim_{x \to 7} \frac{x^3 - 343}{x - 7} = 147).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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