Determine limx →1{(x^2+3 x-4)/(x^2-7 x+6)} The first step is to substitute x=1 into both numerat

step 1 So for this, first 4 x equals 1. So this becomes 1 squared plus 3 into 1 minus 4 upon 1 square m

Determine limx →1{(x^2+3 x-4)/(x^2-7 x+6)} The first step is...

Determine $\lim _{x \rightarrow 1}\left\{\frac{x^{2}+3 x-4}{x^{2}-7 x+6}\right\}$ The first step is to substitute $x=1$ into both numerator and denominator. In this case we obtain $\frac{0}{0}$. It is only when we obtain such a result that we then use L'Hôpital's rule. Hence applying L'Hôpital's rule, $$ \lim _{x \rightarrow 1}\left\{\frac{x^{2}+3 x-4}{x^{2}-7 x+6}\right\}=\lim _{x \rightarrow 1}\left\{\frac{2 x+3}{2 x-7}\right\} $$ i.e. both numerator and denominator have been differentiated $$ =\frac{5}{-5}=-1 $$

Determine $\lim _{x \rightarrow 1}\left\{\frac{x^{2}+3 x-4}{x^{2}-7 x+6}\right\}$
The first step is to substitute $x=1$ into both numerator and denominator. In this case we obtain $\frac{0}{0}$. It is only when we obtain such a result that we then use L'Hôpital's rule. Hence applying L'Hôpital's rule,
$$
\lim _{x \rightarrow 1}\left\{\frac{x^{2}+3 x-4}{x^{2}-7 x+6}\right\}=\lim _{x \rightarrow 1}\left\{\frac{2 x+3}{2 x-7}\right\}
$$
i.e. both numerator and denominator have been differentiated
$$
=\frac{5}{-5}=-1
$$

Key Concepts

Limit

A limit in calculus describes the value that a function approaches as the input (or variable) approaches some particular value. The concept of a limit is fundamental in defining derivatives and integrals, serving as the basis for understanding continuous behavior in functions.

Indeterminate Forms

Indeterminate forms occur in limit evaluations when substitution yields an expression that doesn't clearly define a unique result, such as 0/0. Recognizing an indeterminate form is crucial as it signals the need for further manipulations, such as applying algebraic techniques or calculus tools, to correctly evaluate the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that initially yield an indeterminate form like 0/0 or ?/?. It states that one can take the derivative of the numerator and the derivative of the denominator separately, and then compute the limit of the resulting fraction, provided the conditions for the rule are met.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's output changes with respect to changes in its input. In the context of evaluating limits, particularly when applying L'Hôpital's Rule, differentiation helps in transforming an indeterminate form into a determinate one, enabling the proper evaluation of the limit.

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