Question

Evaluate each limit. $$\lim _{h \rightarrow 0} \frac{a+b+c}{b+c-5}$$

Short answer

The limit is \(\frac{a+b+c}{b+c-5}\), as the expression is constant with respect to 'h'.

Step-by-step solution

Identify the Limit Expression

The given limit is \[\lim _{h \rightarrow 0} \frac{a+b+c}{b+c-5}\]. Since the variable approaching zero is 'h', and 'h' does not appear in the fraction, we can evaluate the limit by inspection.

Evaluate the Limit

The expression \[\frac{a+b+c}{b+c-5}\] does not contain the variable 'h', so the value of the limit is simply the value of the expression itself because the result does not change as 'h' approaches zero.

Key concepts

Limit of a Function

Understanding the concept of a limit is a foundational element in the study of calculus. In simple terms, a limit is a value that a function approaches as the input (or the independent variable) approaches some value.

In the context of the given exercise, \[\lim _{h \rightarrow 0} \frac{a+b+c}{b+c-5}\], the limit is being evaluated as the variable ‘h’ approaches 0. However, a critical observation is that 'h' does not actually appear in the fraction. This implies the output of the function doesn’t depend on 'h' at all; hence, as 'h' approaches zero, the function's value remains unchanged.

Grasping this concept helps avoid confusion when variables unrelated to the limit's approaching value are involved. Essentially, if the variable does not affect the function's value, the limit is the function's value for any substitution of that variable.

Calculus

Calculus is a vast field of mathematics that deals with change and motion. It's divided primarily into two branches: differential calculus, which concerns the rate of change and slopes of curves, and integral calculus, which concerns the accumulation of quantities and the areas under and between curves.

The concept of limits is deeply ingrained in both branches. It not only defines the very concept of a derivative, which measures the rate of change, but also integrates, which sum up areas and quantities. For students, it's important to view limits as the bridge that connects algebra and the dynamic world of calculus, providing a means to quantify and understand the behavior of functions at specific points or as they tend to infinity.

Limit Evaluation Techniques

There are several techniques for evaluating limits, and choosing the right one is essential for finding the correct value efficiently. The most common techniques include:

• Direct Substitution: If a function is continuous at a point, you can find the limit by directly substituting the value into the function.
• Factoring: Limits involving polynomials can often be evaluated by factoring and canceling common factors.
• Rationalizing: When dealing with radical expressions, multiplying by a conjugate may help simplify the expression.
• L'Hopital's Rule: When limits result in an indeterminate form like 0/0 or infinity/infinity, L'Hopital's Rule applies the derivatives of the numerator and the denominator separately.
• Squeeze Theorem: A less often used but powerful method when a function is 'squeezed' between two functions whose limits are known and equal.

For the exercise at hand, because the variable 'h' does not appear in the expression, the best technique is direct inspection - essentially a form of direct substitution where the absence of 'h' makes the function's value independent of 'h's approach to zero.