Evaluate the limit: `lim_{x->-7}{x^2+13x+42}/{x+7}`

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To evaluate this limit, we see that with direct substitution the limit is 0/0, which means that we need to use another method.

In this case, factoring is needed.

`lim_{x->-7}{x^2+13x+42}/{x+7}` factor the numerator

`=lim_{x->-7}{(x+6)(x+7)}/{x+7}` cancel the common factors

`=lim_{x->-7}(x+6)` sub in the value

`=-7+6`

`=-1`

**The limit evaluates to -1.**

The limit `lim_(x-> -7) (x^2 + 13x + 42)/(x + 7)` is required.

If we substitute x = -7 in `(x^2 + 13x + 42)/(x + 7)` the result is `(49 - 91 + 42)/(-7 +7) = 0/0` . The result is indeterminate and in a form that allows the use of l'Hospital's law to determine the limit. Substitute the numerator and denominator by their derivatives.

`(x^2 + 13x + 42)' = 2x + 13`

`(x + 7)' = 1`

The given limit can be written as:

`lim_(x->-7) (2x + 13)/1`

Substituting x = -7 gives:

`(-7*2 + 13)/2 = -1`

The required limit `lim_(x-> -7) (x^2 + 13x + 42)/(x + 7) = -1`

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